FST Name: Functions, Statistics, and Trigonometry Review Unit I Basic Algebra The following pages include practice problems (assignments) that are designed to help you review previously learned concepts. There are a few cases where new information is presented, but these are only slight extensions and/or applications of previously learned concepts. For these topics there are no lesson notes. If you need to review these concepts, please see your teacher and he/she will be able to guide you though a review. Good luck this year and remember to always ask for help when you need it. The material that follows presents an opportunity for LEARNING. Be sure to use it that way! 1 FST Order of Operations Review Name: For a fun game that will help you practice order of operations go to the following website: http://www.mathsisfun.com/games/broken-calculator.html 2 FST Evaluating Expressions Review Name: For 22-25, use w = 5, x = 3, y = -2, and z = 4. 22. xy(5xyz + 5xy) 23. (3xy + 2z)(xz + y)2 3 FST Distributive Property Review Name: 4 FST Multiplying Binomials Review Name: Remember FOIL. It is really using the Distributive Property twice. Ex: (4x – 1)(2x + 5) = Complete the multiplication. 5 FST Solving Linear Equations Practice Name: 1. Solve the following with one step. 1 1 = 2 2 a. x + 3 = -7 b. –4 = x + 4 c. x - x = ____ x = ____ x = ____ d. 12x = 60 e. –40 = -8x f. x= ____ x = ____ x = ____ g. x =5 4 x = ____ h. 1 x = 12 4 i. x = ____ 3 x = -2 4 x = -4 7 x = ____ 2. Solve the following with two steps. 3 x–3 4 a. 3x + 9 = 39 b. –22 = 7x – 8 c. 9 = x = ____ x = ____ x = ____ d. 1 x–3=8 4 x = ____ e. x + .5 = 9.4 3 f. –7 = -2x + 3 x = ____ x = ____ 3. Use the distributive property and then continue solving. 1 (4x + 10) = 16 2 a. 2(x – 4) = 3 b. x = ____ x = ____ c. 5( 4 - 3x) = 15 5 x = ____ 6 FST Name: 4. Solve without using the distributive property. 2 (x – 6) = 10 3 a. 3(x – 5) = 12 b. c. 18 = (6 – 2x)3 x = ____ x = ____ x = ____ a. 5x + 2 = 4x + 6 b. 6x – 3 = 2x – 5 c. –5x – 21 = -2x – 3 x = ____ x = ____ x = ____ 5. Solve using multiple steps. d. 12 = 15 + x -x 4 x = ____ e. x = 18 – 4x 2 f. x = ____ 5 2 x–1= x 6 3 x = ____ 6. Write equivalent expressions without parentheses. a. 8x – (3x + 5) b. 17x – (5x – 9) c. (15x + 7) + (x + 4) _________________ _________________ ___________________ d. (5x + 8) – (2x – 7) e. (7x – 9) – (3 – 2x) f. –(6 – 9x) – ( 11 + x) ________________ _________________ ___________________ 7 FST Name: 7. Solve. a. (5x + 3) + (2x – 4) = 0 b. (2x – 6) – (8x + 4) = 7 x = ____ x = ____ c. 3x + (4 – 7x) = 15 d. 3 – (4x + 1) = 7 + (3 + 2x) x = ____ x = ____ e. 45(x + 6) = 35x – 10 f. 3(x + 7) = 2(x + 9) x = ____ x = ____ g. –1(-x + 5) = 2(x + 6) h. x = ____ x = ____ 1 1 5 1 x + 2( x - ) = 3 3 6 6 8 FST Graphing Name: In this mini-lesson we’re going to review your graphing skills. Graphing is usually not something students enjoy and consequently students do not do justice to the graphs they make. Graphs should be readable. An equation should be recognizable from the position and shape of a graph and vice versa. Please pay attention to your graphing technique. Below are some tips. 1. 2. 3. 4. 5. 6. Label your scale if it is not a scale of 1. Plot actual points that guide your graphing. If there is more than one line or curve on the same coordinate grid, label each with the equation. If axes are hand-drawn, use a straight-edge. For applications, label the axes with the quantities in the relationship. Lines should be straight and curves should be drawn carefully. Graphing Lines These are your options for graphing a straight line. 1. _________________________________ 2. __________________________________ 3. _________________________________ 4. __________________________________ NEATLY graph the following linear equations. Always use a straight-edge when graphing a straight line. Clearly show your method of choice. 1. y = 5x – 4 2. 5y + x = 10 3. –10x + 2y = 12 9 FST Graphing Quadratics Name: Graphs of quadratics are called parabolas. You should plot at least 5 points for every parabola. 1. The vertex 2. The x-intercepts (if they exist; if not plot 2 other points) 3. At least two other points Note that it is usually a good idea to determine these points before you draw your coordinate axes. To find the vertex on your graphing calculator: 1. Enter the equation into your y = editor and graph it. 2. Press 2nd TRACE. (This is the “Calculate” function.) 3. Choose # 3 for MINIMUM (parabola opens up) or # 4 for MAXIMUM (parabola opens down). 4. Left Bound? Move the cursor just to the left of the vertex (using the left/right arrow keys). 5. Right Bound? Move the cursor just to the right of the vertex (using the right arrow key). 6. Guess? Just press enter. The coordinates of the vertex are displayed on the bottom of the screen. To find any x-intercepts on your graphing calculator (you’ll need to find them separately): 1. Enter the equation into your y = editor and graph it. 2. Press 2nd TRACE. 3. Choose # 2 for ZERO. 4. Left Bound? Move the cursor just to the left of the vertex (using the left/right arrow keys). 5. Right Bound? Move the cursor just to the right of the vertex (using the right arrow key). 6. Guess? Just press enter. The coordinates of the x-intercept are displayed on the bottom of the screen. To find 2 more points on your graphing calculator, use the TABLE (2nd GRAPH) after you’ve entered the equation into your y = editor. NEATLY graph the following. Label or list the important points mentioned above. 4. y = x2 – 3x + 2 5. 1 + 10 y = 2x2 – x 3 FST Symbolic Manipulation Review Practice Test Name: Substitute the given value of the variable(s) in each expression and evaluate. 1. 15(2 x 7) x 2 when x 2 1. _______ 34 5 p when p 2 and q 4 2q 2. 2. _______ 3. x – 5 x when x 4 3. _______ 4. (2xy +3z)(xz + y)2 when x = -3, y = 2, and z = -1 4. _______ Simplify each expression. 5. 3(1 11)2 3 5 5. _______ 25 5 2 32 1 16 23 6. _______ 6. 7. 2(1 4) (3) 42 3 7. _______ Expand and simplify each product. 8. –4(5x – 2y –z) 8. __________________ 9. 5 + 8(2x – 4) 9. __________________ 10. 3x – (4x – 5)2 10. __________________ 12 x 3 y 8 z 3 11. __________________ 12. ( x 5)( x 5) 12. __________________ 13. (12 x)( x 5) 13. _________________ 14. (2 x 3) 2 14. _________________ 11. 15. (2x – 4y)2 15. 11 FST Name: Functions, Statistics, and Trigonometry Review Unit II Quadratics The following pages include lesson notes and practice problems (assignments) that are designed to help you review previously learned concepts. There are a few cases where new information is presented, but these are only slight extensions and/or applications of previously learned concepts. You should be sure to have the lesson notes handy during class discussions and fill them out as we work through them. You are also encouraged to add your own ideas to the notes, especially if you understand something in a different way than what is presented here. The lesson notes for the entire unit come first and are followed by the practice problems (assignments). You should be sure to try to complete all the practice problems. In addition, try to stay organized throughout the year. It will help you immensely. Again remember to always ask for help when you need it. The material that follows presents an opportunity for LEARNING. Be sure to use it that way! 12 FST Solving Quadratic Equations By Graphing (Notes) Name: Now that we’ve mastered making “neat” graphs, we’re going to use those skills to help us solve equations. Let’s try to solve x2 – 4 = 0 by graphing. On the right, sketch a “neat” graph of the quadratic function y = x2 – 4. At what points does the graph cross the x-axis? At _______ & _______. Now, substitute the two x-coordinates (separately) into y = x2 – 4. What did you get for the y-value in both cases? _________ Complete the following sentence: To solve an equation that’s set equal to 0 by graphing, graph the equation and look for the _______________________. This is all good so long as the equation is equal to zero to begin with. But, what if it’s 10 = x2 – 6? A little thinking gives us a solution at x = 4 (can you explain why?). But, where does this show up on the graph? Graph y = x2 – 6 in the grid on the right. What are the x-intercepts? _________ and _________. Are these x-values solutions to 10 = x2 – 6? ________. Where do you need to look to find the solutions on the graph? _________________________________________________ Is there a second solution? If so, where is it? ____________ _________________________________________________ If an equation is not set equal to 0, how can you use a graph to find the solutions to an equation? ___________ _________________________________________________________________________________________ _________________________________________________________________________________________ 13 FST Name: Many people don’t like to find solutions this way. Instead they try to change the equation so it is set equal to 0. To the right, show how you could do this for 10 = x2 – 6. Once the equation is equal to 0, you can graph it and find the x-intercepts. You should get the same solution as before. (Try it to make sure it works!) So getting an equation equal to 0 is an important skill. In fact, once you get a quadratic equation equal to 0, and have the exponents written in decreasing order the equation is said to be in standard form. A quadratic equation in standard form looks like this: ax2 + bx + c = 0. Practice putting the following equations in to standard form. a. 5x – 6x2 = 4x2 + 3 b. 3 + 7x + 8x2 = 4x2 – 16x + 5 OK. We now know how to put an equation in standard form and if a quadratic equation is in standard form, the solutions are the x-intercepts. But how many x-intercepts are possible when you graph a quadratic (a parabola)? Hopefully you can visualize that it is possible to have 2, only 1, or even 0 x-intercepts. Below, sketch 3 different graphs with each having the indicated number of x-intercepts (solutions). 2 Solutions 1 Solution 14 0 Solutions FST Name: Now that we know how to find solutions to quadratic equations, we need to see how or why we’d ever want to use this knowledge. After studying the example below, can you think of a situation that is meaningful to you that would require this knowledge? Applications of Quadratics Example: The space shuttle has just landed on the rim of a crater on the earth’s moon. Suddenly, the ground gives way and the shuttle hurtles toward the floor of the crater, 270 feet below. The commander must fire the retrorockets before the shuttle crashes. How much time does he have to do this? 1 2 at + vt + s models the height of a falling object. Fill 2 in the blanks below to describe what each variable represents. The equation h = h = ______________________________________________ t = ______________________________________________ a = ______________________________________________ (On the moon a = -5.4 ft/sec2.) v = ______________________________________________ s = ______________________________________________ If we want to know how long the commander has to fire the rockets, what variable is the unknown? _________ If the space shuttle is just sitting on the moon before the crater gives way, what does v equal? ______________ If the bottom of the crater is 0 feet what should s equal? ____________ What should h equal? _____________ Substitute all of these values into the equation and then enter it on your graphing calculator. Then, use the graph to find the solutions. Solutions: ______________ and _______________ Which of these two solutions answers the question? __________________________________________ Why does the other solution not make sense? __________________________________________ If the floor of the crater were 1080 feet down, how much time would the commander have to fire the retrorockets? ____________________________________________ 15 FST Solving Quadratic Equations Using the Square Root Method (Notes) Quadratic equations in standard form look like this: Name: ax2 + bx + c = 0 We learned from graphing that this type of equation may have 0, 1 or 2 solutions. The solutions you see on the graph are the real number solutions. In reality a quadratic always has 2 solutions, but the solutions may be real numbers or complex numbers (remember i?). To determine how many real number solutions a quadratic equation has without graphing, you can evaluate the _____________________, which is equal to ___________________. If the discriminant is positive, there are ______ real solutions. If it is equal to _______, there is _______ real solution (a double root). If it is ______________, there are no real solutions. How many solutions would you expect from the following equations? Equation 2x + 5x + 3 = 0 4x2 – 28x + 49 = 0 X2 + x + 5 = 0 Value of b2 – 4ac Number of solutions 2 Solving Quadratics Using the Square Root Method In the past, you learned several different methods for solving quadratic equations. They are: 1. Graphing (previous lesson) 2. Square roots (this lesson) 3. Factoring (next lesson) 4. Quadratic formula (last lesson) An equation of the form ax2 – c = 0 can be solved easily by using the Square Root Method. In the 2 examples below, first find the value of the discriminant. Then, solve the equation using the square root method. Example #1: 3x2 – 27 = 0 Example #2: 2x2 + 72 = 0 What happened in Example #2? What does that mean about the type(s) of solutions you get? When is the square root method the most efficient way to solve a quadratic? ____________________________ 16 FST Solving Quadratic Equations by Factoring (Notes) Name: Quadratic expressions are the product of two binomials. Factoring a quadratic means breaking the quadratic back into its binomial parts. Factoring might as well be called LIOF; it’s FOIL in reverse. For starters, let’s say that y = (x + m)(x + n) is the factored form of a quadratic function, where m and n are constants. Using FOIL, we get the general form: y = (x + m)(x +n) y = x2 + mx + nx + mn OR y = x2 + (m + n)x + mn Ex: y = (x + 3)(x + 7) m = 3 n = 7 y = x2 + 3x + 7x + 21 y = x2 + 10x + 21 Do you see how the coefficient of the middle term is the sum m n and the constant is the product m n ? We use that knowledge to work from general form to factored form. Ex: y = x2 + 8x + 15 y = (x + 3)(x + 5) FOIL: y = x2 + 5x + 3x + 15 y = x2 + 8x + 15 Step 1: Factor “15” 1 + 15 = 16 8 Step 2: Add the factors 3 + 5 = 8 Step 3: Set up binomials (x + ___)(x + ___) Step 4: Fill in with factors (x + 3)(x + 5) Examples: Factor the following. 1. y = x2 + 3x – 28 2. y = x2 – x – 42 3. y = x2 – 121 Factoring Out a Common Monomial At times, each term in a trinomial may have a common term that can be divided out to make a new factor. Then you would proceed to factoring the remaining trinomial. Take a good look at the 3 terms in the trinomial. Do you see a number, variable, or both that is contained in each term in the trinomial? If so: 1. Divide each term by that common factor 2. Write the common factor in front of the parentheses containing the new trinomial. Example #1: 6x3 - 42x2 + 60x Example #2: -4x3 + 24x2 + 108x Once you have factored out the common term, the remaining trinomial may be factored. 17 FST Name: More Difficult Factoring Problems All of quadratics we’ve factored so far had one thing in common—the leading coefficient was equal to 1. What happens when this isn’t the case? To factoring a quadratic trinomial of the form: y = ax2 + bx + c with “a” 1 use the “Box Method.” Example #1: Factor y = 2x2 + x - 6. Start by entering the 1st and 3rd terms into the boxes as shown. 2x2 Then, multiply a and c. 2 -6 = -12 -6 Factor -12 so that the sum of the factors equals b, which, in this case, is “1”. -3 4 = -12 and -3 + 4 = 1 Enter the new terms, -3x and 4x, into the free boxes. It doesn’t matter which term goes in each box. 2x2 -3x Now factor all pairs of terms, top to bottom and side to side. 4x -6 x 2x -3 2x2 -3x Write the factors from the top of the box and the side of the box. +2 4x -6 y = (2x – 3)(x + 2) Example #2: Write in factored form. b. y = 5x2 – 13x + 6 a. y = 4x2 - 19x + 12 Solving using Factoring Once the quadratic equation has been factored, the solutions are found by setting each factor equal to zero and solving. Ex: 2x2 – x – 15 = 0 Ex: 6x2 + 2x – 4 = 0 18 FST Solving Quadratic Equations with the Quadratic Formula Name: The Quadratic Formula is used to solve a quadratic equation when other methods (factoring) won’t work. The quadratic formula will solve any quadratic equation, but other methods (factoring), may require less work and should be used when possible. QUADRATIC FORMULA: x b b2 4ac 2a Quadratic equations must be in general form, 0 = ax2 + bx + c to use the quadratic formula because the coefficients a, b, and c must be identified. These values are just substituted into the formula and order of operations are used to evaluate. Example #1: Find a, b, and c for: 10x2 - 13x - 3 = 0 a = ________ b = ________ c = ________ When the Quadratic Formula Goes CRAZY! The graph of y = x2 + x + 2.5 has no x-intercepts. (Graph it, if you don’t believe me!) If you use the quadratic formula to attempt to find the solutions, you get x The numbers 1 12 4(1)( 2.5) 1 9 = 2 2(1) 1 9 1 9 and are not real numbers. They are COMPLEX NUMBERS. 2 2 These types of numbers contain a real component and an imaginary component. The imaginary component contains the number i which is equal to 1 . 1 9 1 i 9 1 3i = = 2 2 2 1 9 1 i 9 1 3i = = 2 2 2 In both of these solutions, -½ is the real component and 3i is the imaginary component. 2 Example #2: Use the Quadratic Formula to find the solutions to the quadratic equations below. Your solutions may be complex numbers. a. x2 - 4x + 7 = 0 b. 9x2 - 4x = -5 19 FST Name: When using the quadratic formula to solve a quadratic equation, your answers should be exact. If b 2 4ac is a perfect square, you should simplify it and continue evaluating the expression. If it is not a perfect square, you may have to leave your answer with the radical intact or simplified. 57 2 57 x = 4 2 Ex: x 4 Ex: x 5 49 2 57 = 7.549834435. . . which is not a perfect square, so leave your answer as is: and 57 . 2 x = 4 49 = 7 5+ 7 = 8.5 2 and 5- 7 = 1.5 2 x = 8.5 and 1.5 Ex: Sometimes answers can be simplified, but not to a single number. x 6 12 2 12 = 3.464101615. . . so we can’t simplify completely. But since 12 and the denominator 2 have a common factor of “2”, we want to simplify so that we can reduce the fractions. x 62 3 6 2 3 becomes x 2 2 2 x 3 3 and x 6 2 3 2 2 x 3 3 20 12 to 2 3 , FST Projectile (Vertical) Motion Formula 1 h at 2 vt s 2 Name: h = ___________________ t = ___________ s = ___________________ a = ___________________ v = ____________________ This formula calculates the height of an object rising or falling under the influence of gravity. Notice the “t2” indicates the equation represents a quadratic and the graph will be a parabola. Nora hits a softball straight up at a speed of 120 ft/s. Her bat contacts the ball 3 feet above the ground. The questions you are expected to answer are usually: 1. What is the maximum height of the object? This is the y-value of the vertex. You can use your graphing calculator to find this. 2. What is the height after t seconds? This answer can be found by two ways. -Substitute the number of seconds t into the equation and evaluate for h. -Enter the equation on your graphing calculator and find the answer in the table. 3. When does the object reach the ground? -This is the x-value when y = 0. You can use your graphing calculator to find this. More examples: 4. In a carnival contest of strength, contestants strike a pad with a hammer. The force of the blow propels a ringer upward toward a bell. If the bell rings, the contestant wins. Marcel swings the hammer and knocks the ringer upward from ground level at 10 m/sec. a. Write an equation to model the situation. b. How high will the ringer be after 0.5 seconds? c. Will Marcel ring the bell 6 meters above the ground? Explain. 21 FST Name: 5. Imagine an arrow is shot from the bottom of the well. It passes the ground level at 1.1 seconds and lands on the ground at 4.7 seconds. a. Write the equation. b. What was the initial velocity of the arrow in meters per second? c. How deep was the well in meters? 6. A ball is dropped from a height of 1000 feet. One second later a second ball is dropped from a height of 750 feet. Which ball hits the ground first? Show or explain how you found your answer. 7. A rock is thrown upward from the edge of a 50 m cliff overlooking Lake Superior, with an initial velocity of 17.2 m/sec. a. Write the equation. b. What is the maximum height of the rock? c. When does the rock hit the water below? 22 FST Name: Solving Quadratic Equations by Graphing (Practice) Use a graph to solve the following quadratic equations. Be sure to include a NEAT graph with the solutions labeled. 1. y = x2 + 2x – 8 2. y = x2 + 2x + 1 3. y = x2 + 2x – 3 x = _____________ x = ______________ x = _____________ 4. y = x2 + 6x 5. y = x2 + 6x + 9 6. y = x2 + 6x – 12 x = _____________ x = ______________ x = _______________ 7. –2 = 4 – 5x + x2 8. 0 = 4 – 5x + x2 9. –10 = 4 – 5x + x2 x = ______________ 23 x = ______________ x = ______________ FST Name: 10. Rewrite each equation in standard form. a. 56 = 56 – 3x + x2 b. 3x + 4 = 4x2 – 1 Use the vertical motion formula h = c. 2x2 – 4 = -2x2 – 6 + 3x d. 1 – 9x + 2x2 = 3x 1 2 at + vt + s, to model and solve the following. 2 Recall: h is the height of the object t seconds after being released, a is the acceleration of gravity (In metric units, a = -9.8 m/sec2; in traditional units, a = -32 ft/sec2), v is upward velocity, t is time in seconds, and s is the object’s starting height. 9. A baseball is thrown straight up with a speed of 80 ft/sec from a starting height of 3 feet above the ground. a. Write the ball’s height as a function of time. _________________________ b. Graph the function on your graphing calculator and then accurately transfer the graph to this paper. c. When will the ball hit the ground? ____________________ 10. An arrow is shot straight up with a speed of 20 m/sec from a starting height 2 meters above the ground. a. Write the arrow’s height as a function of time. _________________________ b. Graph the function accurately. c. When will the arrow hit the ground? ____________________ 24 FST Solving Quadratic Equations using Square Roots (Practice) Name: Solve the following quadratic equations using the Square Root Method. Exact values only. Simplify radicals if necessary. 1. 5x2 – 10 = 0 2. x2 + 81 = 0 3. 2x2 – 3 = 35 4. x2 – 5 = 20 5. 3x2 = -27 6. 3x2 + 87 = -12 7. x2 – 9 = 119 8. x2 + 47 = -800 9. 2x2 + 45 = -445 Use the discriminant to determine the number of real solutions each quadratic equation would have. Equation Value of discriminant (show set-up) Number of solutions 10. 9x2 – 12x + 3 = 0 11. x2 – 2x + 5 = 0 12. x2 + 5x – 6 = 0 13. x2 + 6x + 9 = 0 14. 2x2 + 2x + 3 = 0 15. x2 = -25 16. What does the discriminant of a quadratic equation tell you about the graph of the related quadratic function? __________________________________________________________________________________________ 25 FST Name: Solving Quadratic Equations with Factoring (Practice) Solve each Quadratic Equation using factoring (unfoiling or Box Method). Show all work. 1. 0 = x3 + 5x2 + 6x 2. 0 = 2x3 - 8x2 + 8x 3. 0 = x2 - 2x - 8 4. 0 = 10x2 + 5x - 5 5. 0 = 2x3 - 7x2 + 6x 6. 0 = 40x4 - 10x2 7. 0 = 49x2 - 42xy + 9y2 8. 0 = x2 - 256 10. 0 = 3x2 – 5x3 – 2x4 9. 0 = 30x2 + x - 1 26 FST Factor using the “Box Method.” Then solve. Show ALL work. 11. 0 = 2x2 – 4x – 16 13. 0 = –x2 + x + 6 12. 0 = –6x2 + 15x + 36 14. 0 = 5x2 + 8x – 4 27 Name: FST Solving Quadratic Equations with the Quadratic Formula (Practice) Name: Solve. Show all work. Don’t forget to put in standard form if necessary. 1. x2 + 4x + 49 = 0 2. x2 –5x - 11 = 0 3. x2 – 2x + 2 = 0 4. 2x2 + 6w + 5 = 0 5. x2 = 4x + 9 6. x2 + 22 = 10x 7. x2 – 6x + 11.25 = 0 8. 9x2 + 30x + 24 = 0 9. –7 – 3x2 = 5x 10. 9x2 – 12x + 229 = 0 28 FST Solving Quadratics Test Review Name: Part I—Solve the following equations. Show all work! Exact answers only!! 1 x=6 2 2. 2x – 7 = 13 3. –3x – 5 = 22 4. 3x + 5 = x – 17 5. 2(x + 3) = -4 6. -2(x – 1) + 4 = 5(3x –7) 7. 3x2 = 48 8. 2x2 – 7 = 43 9. x2 + 7 = -43 1. Part II—Solve the following equations by factoring. 10. x2 – 3x + 2 = 0 12. 12x2 -13x – 4 = 0 11. 2x2 + 2x = 4 Part III—Solve the following equations using the quadratic formula. 14. 2x2 – 3x – 2 = 0 13. x2 + 4x + 3 = 0 29 FST Name: 16. 3x2 = -11x – 4 15. x2 + 100x + 30 = 0 Part IV—Make a “neat” graph and use it to help solve the following equations. Round your answers to the nearest hundredth if necessary. 17. x2 + 4x – 32 = 0 x = ______ & ______ 19. x2 – 8x + 19 = 0 18. 6x2 + 17x – 14 = 0 x = ______ & ______ 20. h = -16t2 + 10t + 5 is the equation for an object being launched with an upward velocity of 10 ft/sec2 and starting height of 5 feet. When will the object fall to the ground? \ ___________ 30 FST Name: Functions, Statistics, and Trigonometry Unit 1 Polynomial Functions The following pages include lesson notes and practice problems (assignments) that are designed to help you learn this new concept. In most cases these concepts will be new to you so be sure to follow along and ask good questions. You should be sure to have the lesson notes handy during class discussions and fill them out as we work through them. You are also encouraged to add your own ideas to the notes, especially if you understand something in a different way than what is presented here. The lesson notes for the entire unit come first and are followed by the practice problems (assignments). You should be sure to try to complete all the practice problems. In addition, continue to try to stay organized. It will help you immensely. Again remember to always ask for help when you need it. The material that follows presents an opportunity for LEARNING. Be sure to use it that way! 31 FST Functions and Their Graphs Name: In the real world there are many examples of quantities that are related to each other. Often one of these quantities is dependent on or determined by the other. For example, the number of hours of daylight is dependent on time (the day of the year), the amount of money earned is dependent on number of hours worked, and the cost of sending a package in the mail is dependent on its weight. Give an example of two quantities that are related to each other. _____________________________________ _________________________________________________________________________________________ In the examples above, the quantity that depends on the other is the dependent variable and the other quantity is the independent variable. In the example you gave, which quantity depends on the other?______________________________________ There are several ways to describe the relationship between two variables. You are already familiar with these. 1. Use a table or chart to list the two quantities as an ordered pair. This is usually called an x-y table. 2. Use a formula or equation to describe the relationship between the two quantities. This is sometimes called a function. 3. Use a graph to represent the relationship with a picture. This allows readers to quickly determine rates of change in the variables. Keep in mind that all of these methods give the same information, but in a slightly different way. In this lesson we’re going to concentrate on the last two. Example: You already know how to use a formula or equation to draw a graph. Carefully draw the graph of y = ½ x – 2 on the graph to the right. Example: Many of you probably have a job where you work a certain number of hours per week and as a result make a certain amount of money. Furthermore, in any week both of these values could be different. Because of this we use variables to represent these amounts. Let h = # hours worked in a given week Let e = amount of money earned in a given week Which variable is the dependent variable?_________________________________ If you make $11.50 per hour, how would you calculate how much money you make in a week? ____________________________________________________________________________ Write an equation that describes the relationship between the variables.___________________ This particular relationship is called a function. A function is a set of ordered pairs for which there is exactly one dependent variable for every independent variable. In this case, there is one specific amount of money 32 FST Name: earned for each number of hours worked in a given week. In other words, you can’t work for 20 hours per week for two weeks, and earn $200 the first week, but earn $250 the second. We often use the notation, f(x), to represent a function. The ability to look at a graph and get some idea what function was used to generate the graph is a useful skill. We model all kinds of relationships in the real world using graphs and functions. Being able to relate certain types of graphs with certain types of functions will allow you to more easily find models that you can use to represent the relationship between the two variables. The following graphs represent special types of functions. These graphs are sometimes called parent functions because they are part of a family of many different graphs that are generated by shifting, shrinking and/or “flipping” the graphs you see below. Study the graphs carefully—you’ll want to be able to classify functions according to the types listed below. Linear Function: y = x Quadratic Function: y = x2 Cubic function: y = x3 Absolute Value function: y = |x| 33 FST Name: Square Root function: y = x When talking about functions as models for real-life relationships, it is often important to know something about what values make sense for the variables involved. Acceptable values for the independent variable are known as the __________________ of the function. Acceptable values for the dependent variable are known as the __________________ of the function. Describe the domain and range of each of the functions graphed above. 1. Linear: a. Domain:____________________ b. Range:_________________________ 2. Quadratic: a. Domain:____________________ b. Range:_________________________ 3. Cubic: a. Domain:____________________ b. Range:_________________________ 4. Absolute Value: a. Domain:____________________ b. Range:_________________________ 5. Square Root: a. Domain:____________________ b. Range:_________________________ If we translate (shift) or change the scale (stretch) a function, the domain and range may also change. We’ll take a look at what happens when we do this in the next lesson. 34 FST Transformations of Graphs Name: In the previous lesson you reviewed several different types of parent functions. These functions all went through the origin and had simple rules that involved no addition or multiplication. In this lesson you are going to review what happens to the graphs of these parent functions when they are transformed either through the addition of or multiplication by a constant. There are three types of transformations that we are going to review in this lesson. They are: 1. Translations (shifts) 2. Stretches 3. Reflections Investigation: Graph the following functions on the graphs provided. If necessary, use an x-y table or a graphing calculator. Then, answer the questions that follow. a. f(x) = |x| b. f(x) = |x| + 2 d. f(x) = |x – 3| e. f(x) = |x| -1 c. f(x) = |x + 2| 1. When a constant was added to (or subtracted from) the parent function, as in examples b and e, what happened to the graph of the parent function? ______________________________________________________________________________ 35 FST Name: 2. When a constant was added to (or subtracted from) the independent variable, as in examples c and d, what happened to the graph of the parent function? ______________________________________________________________________________ 3. Describe the transformation on the cubic parent function if f(x) = (x – 2)3 + 1. ______________________________________________________________________________ Investigation: Graph the following functions on the graphs provided. Then, answer the questions that follow. a. g(x) = (x)2 b. g(x) = -(x)2 c. g(x) = 3(x)2 d. g(x) = ½ (x)2 e. g(x) = (x – 1)2 f. g(x) = -2(x – 1)2 + 3 1. When the parent function was multiplied by a negative, as in example b, what happened to the graph of the parent function? ___________________________________________________________________________________ 2. When the parent function was multiplied by a positive constant greater than 1, as in example c, what happened to the graph of the parent function? ___________________________________________________________________________________ 36 FST Name: 3. When the parent function was multiplied by a positive constant less than 1, as in example d, what happened to the graph of the parent function? ___________________________________________________________________________________ Putting It All Together If the parent function is f(x) = x, describe how a, b, and c change the graph when g(x) = a(x – b) + c. a:__________________________________________________________________________________ b:__________________________________________________________________________________ c:__________________________________________________________________________________ Examples: 1. Describe the transformation(s) on the graph of f(x) = x 4 - 2. ________________________________________________________________________________ 2. Given the graph to the right, first determine the parent function. Then, write an equation that models that transformed graph. a. Parent function (dotted graph): f(x) = __________________ b. Transformed function (solid graph): g(x) = __________________ 3. Let f(x) = |x|. Write the equation for the function resulting from a reflection over the x-axis, a vertical shift 3 units downward, and a horizontal shift 2 units to the right. g(x) = ___________________________ 37 FST Polynomial Functions Name: In the past two lessons you’ve studied several different types of functions. In this lesson (and for the rest of the chapter) you will be looking more closely at functions known as polynomials. A polynomial is any expression that contains different ___________ that are ____________ together. Each term is a product of numbers, called ___________________ and variables raised to some _____________. When a polynomial is set equal to a second variable, such as y, you have a polynomial _______________. The general form of a polynomial function looks like this: y = anxn + an-1xn-1 + … + a1x1 + a0. Because we’ll run across one-, two-, and three-term polynomials quite often, you’ll want to know the special names these polynomials have. One-term polynomials are usually called:________________________ Two-term polynomials are usually called:_________________________ Three-term polynomials are usually called:_________________________ Knowing the degree of a polynomial function is quite useful as well. Below, explain how you would determine the degree of a polynomial function. You’ve had plenty of practice graphing polynomial functions already. Let’s turn our attention to some interesting relationships between a polynomial function and its graph. Investigation: Use the polynomial function y = x2 + 6x + 8 to complete the following steps. 1. Classify this polynomial function (give its degree and name). __________________________________ 2. Graph this polynomial. List the x-coordinate of the x-intercepts. x-intercepts: __________ and __________ 3. Would it be possible for there to be more than two x-intercepts for this type of function? ___________________________________________________ 4. Factor this polynomial (“unfoil” or use the box method). ___________________________________________________ 38 FST Name: 5. Call the two factors you wrote above (x – r1) and (x – r2). What relationship do the x-intercepts have with r1 and r2 (r1 and r2 are sometimes called roots)? ___________________________________________________ OK. So maybe you remember this from just a few lessons ago. It is SO important however that it was worth doing again. There are a couple of conclusions that you can make from this investigation. 1. The degree of the polynomial is the ____________________________________________________. 2. The x-intercepts of a polynomial determine ______________________________________________. Example: Determine the number of x-intercepts for: y = x2 + 10x + 25. ___________________ You probably thought that there should be two x-intercepts because the polynomial is degree two. If you did, good for you. But hold on a second! Try graphing the polynomial to the right. Then factor it below. How many x-intercepts were there? _____________________ Where was the x-intercept located? _____________________ How did this show up in the factorization of the polynomial? ________________________________________ Try graphing y = x2 + 10x + 30 on your graphing calculator. How many x-intercepts are there? ____________ Frustrated? Don’t be! As it turns out the degree doesn’t determine the exact number of x-intercepts, but it does determine the maximum number of x-intercepts. There can be fewer. In the case where the graph simply touches the x-axis (it doesn’t cross there), the x-intercept can be used as a factor more than once. This was the case for the polynomial function y = x2 + 10x + 25. In the case where the graph doesn’t intersect or touch the x-axis, the factors use imaginary roots. This was the case for y = x2 + 10x + 30. So, in the end, there are always the same number of linear factors for a polynomial function as the degree of the polynomial indicates. Sometimes, the same factor is used more than once. Sometimes the factors contain imaginary roots. In the next lesson, we’ll learn more about how to find the factors of a polynomial function if the x-intercepts aren’t nice numbers. In addition we’ll look at higher-degree polynomials! 39 FST Factoring Polynomials (Part 1) Name: From the last lesson, you already know that the x-intercepts of a polynomial function determine at least some of the linear factors of that function. You also know that for an nth degree polynomial there will be n factors. In this lesson we are going to utilize these facts in a couple of different ways. Example A: A 3rd-degree polynomial function is called a cubic function. At right is a graph of the cubic function: y = -4x3 – 16x2 + 9x + 36 The x-intercepts of the function are –4, -1.5, and 1.5, so its factored equation must be in the form: y = a(x + 4)(x + 1.5)(x – 1.5) To find the value of “a”, you can substitute the coordinates of another point on the curve. The y-intercept is (0, 36). Substitute this point into the equation for x and y, and find the value of “a”. Show your work. So a = _______, and the factored form of the equation is y = ____________________. Let’s try another example. Example B: Write the equation of the function below in factored form, including a value for “a”. Show your work. y = __________________________ This method works well when the zeros of a function are integer values. Unfortunately, this is not always the case. Sometimes the zeros of a polynomial are not “nice” rational or integer values, and sometimes they are not even real numbers. With quadratic equations (degree 2), if you cannot find the zeroes by factoring or making a graph, you can always use the quadratic formula. Once you know the zeros, r1 and r2, you can write the polynomial in the form: y = a(x – r1)(x - r2) 40 FST Name: If the degree of the polynomial is larger than 2, you hope that there are the same number of x-intercepts as the degree of the polynomial and that it is easy to determine the value of those x-intercepts. This was the case in Example A at the beginning of the lesson. But what happens if the number of x-intercepts is not the same as the degree of the polynomial? There are two possible answers. 1. One (or more) of the x-intercepts is counted as a root more than once. 2. One (or more) of the roots is imaginary. In the first case, you’ll know that an x-intercept is counted as a root more than once because it will only touch the x-axis as in Example B. In the second case, you’ll know that one (or more) of the roots are imaginary because there aren’t enough real roots. (Remember the number of roots is always the same as the degree of the polynomial.) You might detest imaginary numbers. But there is actually something pretty cool about them if they serve as the roll of a root. If a + bi is a root, then its complex conjugate a – bi is also a root. Example C: Find a 4th-degree polynomial function with real coefficients and zeros x = -2, x = 3, and x = 1 – i. (Note: You do not have to find a for this problem.) A little later we’ll learn a technique for finding additional roots of a polynomial if only one root is known. I guarantee you’ll love it! In the next lesson, we’re going to review a few important algebraic skills. 41 FST Multiplying Factors Name: In the previous lesson you were asked to find the factored form of a polynomial using the x-intercepts on a graph. You may also be asked to write the equation as a polynomial in standard form. To do this you simply multiply all of the factors together. Let’s take some time to get some more practice doing that. Example 1: Multiply: (x – 12)(x – 3). Example 2: Multiply: x(x +6)(x – 9). Example 3: Multiply: (x – 3)(x – 5)(x + 7). Example 4: Multiply: (x + 1)(x – 2)(x - 5)(x +3) Change the following from factored form to a polynomial in standard form. Show all work. 1. 5(x – 7)(x + 11) 2. –7(x – 10)(x + 10) 3. (x + 5)(x – 3)(x – 7) 4. (x + 4)(x – 5)(x + 1) 42 FST Name: 5. (x – 6)(x + 6)(x – 8) 6. (x – 1)(x – 1)(x + 3)(x + 3) 7. (x + 4)(x – 2)(x + 3)(x + 5) 8. –5(x – 4)(x + 7)(x – 1)(x + 4) Write an equation in standard form for the following polynomials. 9. The y-intercept is at (0, 3). 10. The graphs goes through (-1, 8). 43 FST Factoring Polynomials (Part 2) Name: From previous lessons, we already know that you can find the roots of a quadratic function by: 1. ________________________________ 2. ________________________________ For all polynomials, you can find the roots by looking for the ___________________________ on the graph. Unfortunately these roots may only be approximations, and/or the roots won’t be real numbers. In this lesson you are going to learn a method for finding the exact roots, both real and nonreal, of higher-degree polynomials. You remember the good old days of elementary school, right? Well, let’s take a trip down memory lane and revisit LONG DIVISION!! Example A: Use long division to divide 1024 by 8. Show your work to the right. That was fun! Now we’re going to apply those same skills to dividing polynomials! Study the example below carefully. Then watch as your teacher goes through the process. You may want to do the problem on your own off to the right of the example provided to get some extra practice. Example B: Use long division to divide (x5 – 6x4 + 20x3 – 60x2 + 99x – 54) by (x3 – 6x2 + 11x – 6). The two factors are (x2 + 9) and (x3 – 6x2 + 11x – 6). When you divide polynomials, be sure to write both so that the terms are in order of decreasing degree. If a degree is missing, insert a term with a coefficient 0 as a placeholder as shown in the example below. With this knowledge in hand we can return to our quest of factoring. 44 FST Example: Rewrite y = 9x3 – 31x – 10 in factored form. Name: The first step is to graph the function and look for “nice” x-intercepts. From the graph, we know x = _________ is an x-intercept. This means ________________ is a linear factor. Now we divide. For extra practice, cover up the solution below and do the division on your own to the right. 9 x 2 18 x 5 ( x 2) 9 x3 0 x 2 31x 10 (9 x3 18 x 2 ) 18 x 2 31x (18 x 2 36 x) 5x 10 (5 x 10) 0 So we now know 9x2 + 18x + 5 is a factor of the original polynomial. Better yet, its degree is two. That means we can use the quadratic formula to find its roots! Please show your work for that below. 1 5 So the roots are: 2, , and . This means the linear factors are: ____________________________. 3 3 To find the value of “a”, use the y-intercept as a point. The y-intercept is: _______________________. Show your work on finding the value of “a” below. The final factored form is: ___________________________________________. 45 FST Let’s summarize everything we’ve done until now. Name: To factor a higher-degree polynomial (degree 3 or higher): 1. _________________________________________________________ 2. _________________________________________________________ 3. _________________________________________________________ 4. Repeat this process until your answer in step 3 is a quadratic. 5. _________________________________________________________ Your assignment will give you more practice with this process. However, there is one last thing you need to know. It is called the Factor Theorem. The expression (x – a) is a factor of P(x), if P(a) = 0. In other words if you know a root of a polynomial and substitute that value into the polynomial and simplify, you will get zero for an answer. Because the root causes the polynomial to equal zero, roots are often called zeros. Example: Verify that (x – 3) is a factor of P(x) = x4 – 13x2 + 36. Substitute _______ into P(x). You should get 0. Show that you do here. You can use this simple process to check to make sure your linear factors are correct. It’s always good to check your work now! Extra Practice: Below, show your work to the polynomial division problem your teacher provides to you. 46 FST Synthetic Division Name: Synthetic division is a shortcut method for dividing a polynomial by a linear factor. Example A: Consider this division of a cubic polynomial by a linear factor: 7 x 3 3 x 2 56 x 24 3 x 7 Example B: Divide using synthetic division. Example C: Simplify: 6 x 3 11x 2 17 x 30 x2 3x 3 5 x 2 15 x 25 5 x 3 47 FST The Rational Root Theorem Name: From previous lessons you know that one of the first steps in finding the zeros of a polynomial is to graph the polynomial and identify the x-intercepts. If the x-intercepts of a graph are not integers, identifying the zeros is more difficult. The Rational Root Theorem tells you which rational numbers might be zeros. (Remember rational numbers are just fractions.) Rational Root Theorem: If the polynomial equation P(x) = 0 has rational roots, they are of the form p where p is a factor of the q constant term and q is a factor of the leading coefficient. This theorem helps you narrow down the values that might be zeros of a polynomial function. Note that it only finds rational roots. It won’t find roots that are irrational or that contain imaginary numbers. To find these roots you must use the quadratic formula. This, of course, requires that you have a factor that is degree 2. This is why we spent the last few days learning how to divide polynomials. Once we find a rational root, we divide it into the polynomial with the hope that you will eventually get a factor that is a quadratic. Example: Find the roots of 7x3 – 3x2 – 56x + 24 = 0. Show ALL work. 1. First graph the polynomial using the window below. None of the x-intercepts are integers. Window: 2. Identify the constant value. ____________ Xmin: -4 Xmax: 4 Xscl: 1 Ymin: -80 Ymax: 80 Yscl: 10 3. List its factors. ______________________ These are your “p” values. 4. Identify the leading coefficient. _________ 5. List its factors. ______________________ These are your “q” values. p possibilities: ___________________________________________________________________ q p You know there are no integer roots, so which values can you ignore? CROSS THEM OFF YOUR LIST. q 7. List integer values between which an x-intercept exists (for example, between 2 and 3). 6. List all a. _________________ b. __________________ c.____________________ 8. Do you have any numbers in your list that are between either of these two values? _______________ 48 FST Name: There are no numbers in your list between –3 and –2. This means that the value of that root is irrational. There are numbers on your list between 0 and 1. They are _____________. Which of these comes closest to the actual root? _____________ The factor theorem says that if this value is a root, then when you substitute it into the equation, you will get 0. Verify that this value is a root. Remember that your original polynomial function is: 7x3 – 3x2 – 56x + 24 = 0. 9. Did you get “0”? _________ If so, write the number as a linear factor. ___________ 10. Now use long division to find another factor of your polynomial. Show your work below. 11. The factors are _____________ and __________________________. (x – r1) (The factor you just found by dividing.) We can now find the roots for the polynomial. One of them we already know. The other two we can find using the quadratic formula (since the other factor is degree 2). 12. The roots (zeros) are: __________ ___________ ___________ (3rd degree poly = 3 roots) 49 FST Name: Solving Polynomial Equations Make sure you aren’t confused by the terminology. All of these are the same: Solving a polynomial equation P(x) = 0 Finding roots of a polynomial equation P(x) = 0 Finding zeroes of a polynomial equation P(x) = 0 Factoring a polynomial equation P(x) There’s a factor for every root, and vice versa. (x – r) is a factor if and only if r is a root. This is the Factor Theorem: finding the roots or finding the factors is essentially the same thing. How do you find the factors or zeroes of a polynomial (or the roots of a polynomial equation)? Basically, you chisel. Every time you chip a factor or root off the polynomial, you’re left with a polynomial that is one degree simpler. Use that reduced polynomial to find remaining factors or roots. Try the flowchart below. Solving Polynomial Equations Ask: How many roots should I expect? nth degree = n roots 1st Degree Linear Look at the graph. I can see the xintercept as an integer value. Write it down; you are done. When looking at the graph make sure you can see the local shape of the graph by adjusting your window. The x-intercept is not an integer value. Solve the equation to find the value. You can divide a polynomial by using long division or synthetic division. 2nd Degree Quadratic Look at the graph The graph crosses the x-axis twice at integer values. Write them down; you are done. The graph touches the xaxis in one place at an integer value. This is a double-root. Write it down; you are done. 50 The graph does not go through the x-axis at all or it crosses the x-axis between the integer values. Use the quadratic formula. FST Name: 3rd Degree Cubic Look at the graph The graph crosses the x-axis 3 times at integer values. Write them down; you are done. The graph crosses the xaxis once at an integer value. This is 1 root. It also touches the x-axis at an integer value. This is a double-root. Write them down; you are done. The graph crosses the x-axis once at an integer value. This is 1 real root; write it down. You need 2 complex roots. Divide your cubic by the factor using the first root. This will create a quadratic polynomial. Find the roots of this quadratic by following the directions above. 4th Degree Quartic Look at the graph. The graph crosses the x-axis 4 times at integer values. Write them down; you are done. The graph crosses the x-axis twice at integer values. These are 2 real roots. Write them down. You need 2 complex roots. Two options are below. Make a factor out of one of the real roots you found on the graph. Divide the polynomial by the factor. You will create a cubic polynomial. Find the roots of the cubic by following the directions above. Or Make factors out of both roots. Multiply these two factors together. This will create a quadratic. Divide the given quartic by the quadratic. This will produce a second quadratic, which the quadratic formula will solve. For higher degree polynomials, always look for the nice x-intercepts on the graph. Write them down and turn them into linear factors. Multiply the linear factors together (remember to account for double roots!) and divide the original polynomial by the answer. If you get a degree 2 polynomial after you’ve divided, use the quadratic formula to find the last two roots. 51 FST Functions and Their Graphs Practice Name: Directions: For each of the relationships given, classify the variables as dependent and independent. 1. The height of a child and his/her age. Dependent:____________________ Independent:___________________ 2. Oven temperature and length of time it takes to bake cookies. Dependent:____________________ Independent:___________________ Directions: For each of the functions given, produce a x-y table and then neatly graph the function. Use the integers between –2 and 2 as your domain. 3. f(x) = -2x + 1 4. g(x) = ½ (x + 1)2 – 2 5. h(x) = 2 |x – 3| + 1 6. s(x) = 52 2x 5 FST Name: Directions: Identify the class of function (linear, quadratic, etc.) for each of the graphs below. Then, describe the function’s domain and range. 7. 8. Type:___________________________ Type:___________________________ Domain:_________________________ Domain:_________________________ Range:__________________________ Range:__________________________ 9. Type:__________________________ Domain:________________________ Range:_________________________ 10. Describe a real-life situation that has a relationship between two variables. Classify the each variable as independent or dependent. Then, make a good guess at which type of function (linear, quadratic, etc.) might represent the relationship and why you think this way. 53 FST Transformations of Graphs Practice Name: 1. Let f(x) = |x|. Write the equation of the function resulting from a vertical shift of 3 units downward and a horizontal shift of 2 units to the right. 1.____________________________ 2. Let f(x) = x2. Describe the graph of g(x) = -(x+2)2. 2.____________________________ Directions: For each of the functions given, identify the parent function, the transformation involved, and then neatly graph the function. 3. f(x) = -x2 + 1 4. g(x) = (x + 1)2 – 2 5. h(x) = |x – 3| + 1 6. s(x) = 54 x5 FST Name: Directions: For each graph, identify the parent function, the transformation(s) involved, and then write an equation for the graph. 7. 8. Parent:___________________________ Parent:___________________________ Function:_________________________ Function:_________________________ 9. 10. Parent:___________________________ Parent:___________________________ Function:_________________________ Function:_________________________ (Hint: There is a vertical stretch on this one) 55 FST Polynomial Functions Practice Name: 1. Identify the following polynomials by degree and name. a. 3x2 + 6x – 9 b. 5x6 – 25 ________________________________ __________________________________ 2. Determine the maximum number of x-intercepts for the following polynomial functions. a. y = 2x3 + 4x – 1 b. y = x2 + 12x + 36 ________________________________ __________________________________ 3. Without graphing, provide the x-intercepts of the following polynomial functions. a. y = (x – 2)(x + 3) b. y = (2x – 8)(x + 1) ________________________________ __________________________________ c. y = (x – 7)2 d. y = x(x – 5) ________________________________ __________________________________ 4. Factor the following polynomial functions to find the x-intercepts. a. y = x2 + 12x + 20 b. y = 2x3 –12x2 – 14x ________________________________ __________________________________ 5. Graph the following polynomial functions and determine the x-intercepts. a. y – 24 = x2 + 11x b. y = 2x2 – x – 10 ________________________________ __________________________________ 6. Write the factored form of a polynomial function whose graph contains the following x-intercepts. a. (0, 0) and (-5, 0) b. (2, 0); (-3, 0); and (½, 0) ________________________________ __________________________________ 56 FST Factoring Polynomials (Part 1) Practice Name: 1. Write the factored form of the polynomial that has the x-intercepts and the given point shown. (You must find a.) SHOW YOUR WORK!! a. b. _______________________________ ___________________________________ 2. A certain polynomial has (x – 2 + 3i) as one of its factors. Give another factor for this polynomial. ______________________________ 3. A certain 4th degree polynomial has only one x-intercept at x = -2. If this polynomial also has a zero at x = 3 – 2i, write the factored form of this polynomial. (You do not need to find a). ______________________________ 4. Find the factored form of y = 4x3 + 8x2 – 36x – 72. Include a graph and your work finding “a”. 57 FST Factoring Polynomials (Part 1) More Practice Name: 1. Without graphing, find the x-intercepts and the y-intercept for the graph of each equation. Then, rewrite the equation as a polynomial in standard form. a. y = (x + 6)(x – 5) b. y = -(x – 8)2 c. y = 2(x + 1)(x – 1) d. y = 3(x + 4)(x + 2) e. y = -(x + 2)(x – 1)(x - 6) f. y = 0.75x(x – 2)(x + 6) 2. Write a polynomial function with the given features. a. A quadratic function whose graph has only one x-intercept, -4, and whose y-intercept is –8. __________________________ b. A cubic function with leading coefficient –1 whose graph has x-intercepts 0 and 5, where x = 5 is a double root. __________________________ c. A quadratic function whose graph has vertex (3, -8), which is a minimum, and two x-intercepts, one of which is 5. __________________________ d. A fourth-degree polynomial function with two double roots, 0 and 2, and whose graph contains the point (1, -1). __________________________ 58 FST Name: 3. For the following graphs, neatly list the following: a. All zeroes. b. The y-intercept. c. The lowest possible degree of the function. d. The factored form for each polynomial function. Include the value of a. Check your equation on your graphing calculator. It should create a graph that matches the one given. i. Each tick mark is 1 unit. The graph goes through (1, -4). ii. Each tick mark is 1 unit. iii. Each tick mark on the x-axis is 1 unit. Each tick mark on the y-axis is 5 units. The graph goes through (0, 24). 4. Write the equations for each graph above as a polynomial in standard form. a.______________________ b.______________________ 59 c.______________________ FST Polynomials Quiz Review Name: Answer the following questions. 1. Which point on a quadratic function should you focus on to determine a translation of the parent function? _______________________ 2. Can you name another function for which you would focus on that same point to determine a translation of the parent function? _______________________ 3. What is the domain for any quadratic, or absolute value function? _______________________ 4. How do you determine the range for any quadratic or absolute value function? _______________________ 5. Write the equation for the given transformations: a. Parent: Square Root Transformations: Horizontal shift left 2 units, vertical stretch by a factor of 5. _______________________ b. Parent: Absolute Value Transformations: Horizontal shift right 6 units, vertical shift down 1 unit. _______________________ c. Parent: Quadratic Transformations: Vertical shift up 4 units, horizontal stretch by a factor of ⅔. _______________________ 6. Give the degree of the polynomial 4x7 – 6x6 + 5x4– 12x + 2. _______________________ 7. Give the x-intercepts for the following equations. a. (2x + 5)(x - 2) = 0 b. -x(2x - 5) = 0 c. 0 = 4(x – 1)2 ___________________ ____________________ _______________________ 8. Rewrite the following equations as polynomials in standard (general) form. Show all work! 1 (x – 2)(x + 3) 3 b. y = -3(x + 3)(x + 3)(x – 2)(x – 5) _______________________ ____________________________ a. y = 60 FST Name: c. y = 2 (x – 4)(x + 1)(x – 3) 5 d. y = 2(x – 7)(x – 1)(x + 3)(x – 4) ________________________ _____________________________ 9. Give the parent function and then describe the transformations for: y = 3 (x – 7)2 – 2. 2 Parent function: ______________ Transformations: _________________________________________________________ Show all work in answering the questions below. 10. Solve the equation x2 + 6x + 13 = 0. (Exact values please!) ________________________ 11. Solve the equation x4 – x3 – 2x2 = 0 ________________________ 12. Give the roots of y = -3(x + 7)(x + 4)(x - 2), Then write as a polynomial in standard form. Roots: ________________________ Polynomial:____________________ 61 FST Name: 13. Write an equation for the function graphed. Please find a. ______________________________ 14. Write an equation for the function graphed. Please find a. Hint: One of the zeros occurs at x = 3i. Also, i2 = -1. ______________________________ 15. Write y = 1 3 1 2 x - x – 2x + 3 in factored form. 4 4 16. Find the polynomial function whose graph has x-intercepts -1, -2, and 8, and a y-intercept of -64. ______________________________ ______________________________ 17. Give the complex conjugate for 7-3i. ____________ 18. Write the polynomial function of degree 4 with roots at x = 5 and x = 2 - 4i. Do not determine the value of “a”. ______________________________ 62 FST Long Division Group Work Name: Each group member will do one problem from 1-3 and one problem from 4-6. One person works at the board while one person learns/teaches, and one person writes the problem on the paper. Then rotate. Record who did each job below the problem. 1. x 3 2 x 2 21x 30 x 5 2. Board_____ Learn/Teach_____ Write ______ 3. Board_____ Learn/Teach_____ Write ______ 6 x 3 x 2 26 x 21 3x 7 4. Board_____ Learn/Teach_____ Write ______ 5. 2 x 3 13x 2 29 x 21 2x 3 3x 4 2 x 3 7 x 18 x2 Board_____ Learn/Teach_____ Write ______ 4 x 5 7 x 3 5 x 2 4.5 2x 3 6. Board_____ Learn/Teach_____ Write ______ 4 x 7 3x 5 2 x 2 3 x 1 Board_____ Learn/Teach_____ Write ______ 63 FST More Practice with Polynomial Division Name: Divide. 1. ( x 2) 3x3 8x 2 11x 30 2. ( x 4) x 4 13x 2 48 64 FST Factoring Polynomials Practice (Part 2) Name: 1. Consider the function y = x4 + 3x3 – 11x2 – 3x +10. a. How many zeros does the function have? ________________________ b. Name the zeros. ________________________ c. Write the polynomial function in factored form. (Don’t forget to find a.) ________________________ 2. Consider the function P(x) = 2x3 – x2 +18x – 9. a. Verify that ½ is a zero. b. Find the remaining zeros of the function. 3. Divide using long division. Show all work. a. ( x 2) 3x3 8x 2 11x 30 b. ( x 4) x 4 13x 2 48 65 FST Name: c. (2 x 1) 32 x5 1 d. ( x 3) x3 7 x 2 11x 3 4. Find all roots of the polynomial inside the division sign. Show all work. (Don’t forget that the linear factor has a root and that when you get a quadratic you can use the quadratic formula to find its roots.) a. ( x 2) 3x3 8x 2 11x 30 b. ( x 4) x 4 13x 2 48 66 FST Synthetic Division Practice Name: 1. Find the missing value in each synthetic-division problem. 2. a. b. a = ____ b = ____ c. d. c = ____ d = ____ Divide using synthetic division. Show all work. One of these 4 problems has a remainder. Write your answer as a polynomial. a. 2 x3 3x 2 x 2 x2 b. x 4 13x 2 55 x 25 c. x5 3x 4 2 x 2 15 x 10 x2 x4 1 d. x 1 67 FST Name: 3. Divide using synthetic division. Show all work. Write your answer as a polynomial. There should be no remainders. a. x3 x x 1 b. 6 x 4 5 x 2 11 x 1 c. 4 x3 42 x 2 11x 90 x 10 d. 5 x5 30 x 4 x 6 x6 68 FST More Practice with Polynomial Division Name: 1. Divide using long division. There should be no remainders. Show all work. a. x 2 3x3 8x 2 11x 30 b. x 4 x 4 13x 2 48 c. 2 x 1 32 x5 1 d. x 3 x3 7 x 2 11x 3 2. Divide using synthetic division. Show all work. One of these 4 problems will have a remainder. Write your answer as a polynomial. a. 2 x3 3x 2 x 2 x2 b. 69 3x 4 2 x 2 15 x 10 x2 FST Name: x 23x 4 x 30 x5 4 c. x 1 x 1 4 2 d. 3. Use the Factor Theorem to decide if the given number is a zero of the polynomial. a. Decide if 3 is a zero of P(x) = x3 – 7x2 + 11x + 3. b. Decide if 1 is a zero of P(x) = x3 + 2x2 – 3x + 4. c. Decide if 5 is a zero of P(x) = x4 – 4x3 – 6x2 + 3x + 10. 4. Name the potential root given by the linear factor. Then use synthetic division to verify if that potential root is in fact a root. a. x3 2 x 2 3x 6 x2 b. 70 x 3 3x 2 2 x 11 x4 FST Rational Root Theorem Practice Name: 1. Consider P(x) = 6x4 – 4x3 + 3x2 + 13x – 10. a. List the factors of: -10 _______________ and 6 ________________. b. Give the possible ratios p . __________________________________________________ q c. Find a root of P(x) = 0 from the rational numbers you wrote above. Explain why you chose this root. SHOW ALL WORK! 2. Solve 3x3 + 2x2 + 3x + 2 = 0. Show all steps and work. (Use your notes.) One more on the next page. 71 FST 3. Solve 5x4 – 4x3 + 19x2 – 16x – 4 = 0. Show all steps and work. Name: Hint: In this problem, you can see a “nice” root by looking at the graph. You can start dividing the 4th degree polynomial given above using that root first. Then you will need to use the Rational Root Theorem on the cubic polynomial to find a root for it. Then divide the cubic polynomial by that root. Finally, use the quadratic formula to find the roots of the answer. 72 FST Solving Polynomial Equations Name: Use your flow-chart and knowledge from this chapter to solve each equation. Show all work, including any graphs you use in the process. 1. 3x – 9 = 0 2. x2 – 5x + 6 = 0 3. 3x2 + 9 = 0 4. x2 – 4x + 16 = 0 5. x3 + 2x2 – x – 2 = 0 6. x4 – 16 = 0 7. x3 + x2 – 5x + 3 = 0 73 FST Name: 8. x3 + x2 + 4x + 4 = 0 9. 9x4 + 3x3 – 30x2 + 6x + 12 = 0 10. x4 – 8x3 + 73x2 – 228x + 212 = 0 74 FST Name: 11. 18x3 + 27x2 – 2x – 3 = 0 12. 15x4 + 6x3 – 2x2 + 3x + 77 = 0 75 FST Solving Polynomial Equations Practice Test 1. List the roots of Name: 1 (x + 1)(x – 2)(x - 4) = 0 4 ________________ 2. Write the equation in #1 as a polynomial. Give the degree of this polynomial. _____________________ 3. Solve –x(2x – 1)(x + 9) = 0 _______________ 4. Write an equation for the functions graphed below. Don’t forget to find the value of a. a. b. _______________________ __________________________ 5. Find a polynomial function whose graph has a. x-intercepts at –1, -2, and 8, and the y-intercept at –64. b. x-intercepts at –4, -3, 5i, and –5i, and the y-intercept at 200. ___________________________ ___________________________ 76 FST Name: 6. Find the factored form of the polynomial function P(x) = 3 3 3 2 x - x - 66x + 126. 2 2 Show how you found your answer. ____________________________ 7. Write the factored form of the polynomial graphed right. (Hint: x = -2 is a TRIPLE root.) _________________________________ 8. Determine the degree of the polynomial graphed on the right. Specifically explain how you determined the degree. _______________________________________________ _______________________________________________ 9. Write the lowest degree polynomial function with zeros at 2i and –5i, and y-intercept of 12. You may leave your answer in factored form. _________________________________ 1 10. Is “- ” a possible zero of P(x) = 3x3 – 8x2 – 33x - 10? Show your work. 3 ________ 1 11. Is “ ” a possible root of P(x) = 3x3 + 25x2 + 53x + 15? Show your work. 3 ________ 77 FST Name: 12. Use long division to divide the following. a. ( x 3) 2 x3 3x 2 11x 6 b. ( x 4) x 4 8x3 40 x 2 28x 16 13. Use synthetic division to divide the following. a. x3 5 x 2 25 x 125 x5 ______________________ b. x 4 8 x3 68 x 56 x2 _________________________ 14. Find the roots of 15x4 + x3 – 21x2 – x + 6 = 0. Show all work, including graphs. ___________________ 78 FST Name: 15. Find all roots for P(x) = x3 + 13x - 34 = 0. Show all work, including any graphs. ______________________ 16. Find all roots for P(x) = 48x4 – 76x3 – 148x2 + 101x – 15. Show all work, including any graphs. _______________________ 79 FST Name: Functions, Statistics, and Trigonometry Unit 2 Rational Expressions The following pages include lesson notes and practice problems (assignments) that are designed to help you learn this new concept. In most cases these concepts will be new to you so be sure to follow along and ask good questions. You should be sure to have the lesson notes handy during class discussions and fill them out as we work through them. You are also encouraged to add your own ideas to the notes, especially if you understand something in a different way than what is presented here. The lesson notes for the entire unit come first and are followed by the practice problems (assignments). You should be sure to try to complete all the practice problems. In addition, continue to try to stay organized. It will help you immensely. Again remember to always ask for help when you need it. The material that follows presents an opportunity for LEARNING. Be sure to use it that way! 80 FST Introduction to Rational Functions (Hyperbolas) Name: Over the next few days, you will learn how to: Model real-world data with a rational function Examine transformations of f(x) = 1/x, the parent function for the inverse variation curve Rewrite equations for rational functions to see how they are related to y = 1/x Write an equation for the graph of a rational function Use rational expressions to solve a problem involving acid solutions We’ll start by introducing a new function—the rational function. The investigation that follows will help you become more familiar with the behavior of rational functions. Investigation: “The Breaking Point” For this investigation, work with a partner. If you want to use the data provided on the right instead of collecting your own data, skip to step 8. Step 1: Lay a piece of spaghetti on a table so that its length is perpendicular to the edge of the table. The end of the spaghetti should extend over the edge of the table. Step 2: Measure the length of the spaghetti that extends beyond the edge of the table. Step 3: On a piece of graph paper, make an x-y table. One variable should be length. Record the measurement from step 2 in this column. Step 4: Tie (or tape) the string to the paper cup so that you can hang it from the end of the spaghetti. Step 5: Place pennies (beans, M&M’s, etc.) into the cup one at a time until the spaghetti breaks. Length (cm) 16 16 15 15 14 13 13 12 12 11 10 9 8 7 6 Step 6: Record the number of pennies (beans, M&M’s, etc.) that were in the cup when your spaghetti broke. This value should be recorded in the mass column on your x-y table. Mass (# of pennies) 6 5 7 6 6 7 6 8 7 8 9 10 11 13 16 Step 7: Repeat steps 1-6 at least 5 times before moving onto step 8. Combine your data with the data from at least one other group. Step 8: On your graph paper, make a “neat” graph of your data with length as the independent variable, x, and mass as the dependent variable, y. Which graph on the right, best represents your data? ________ Step 9: Write a few sentences below that describe the relationship between mass and length. 81 FST Step 10: Put your data into L1 and L2 on your calculator (STAT menu; EDIT). Name: Step 11: Use STAT: CALC to do a “PwrReg”. This is Option “A” in the STAT: CALC menu. Write down the equation your calculator gives you with only positive exponents. The relationship between length and mass in the investigation is an inverse variation. The parent function for an inverse variation curve is f(x) = 1/x. This curve is the simplest type of rational function. P( x) where P and Q are both polynomials and the degree of Q(x) 1. Q( x) Rational functions that are hyperbolas have the degree of P(x) = 0. A rational function is written as f(x) = Properties of Hyperbolas The graph of y = 1/x is shown to the right. It is a hyperbola rotated 450 with vertices at (1, 1) and (1, -1). A hyperbola has two asymptotes (lines the graph approaches but does not reach). Give the equation of both asymptotes. x = ___________; y = ____________ The asymptotes for this graph also happen to be the x- and y-axes. Why doesn’t the graph of a hyperbola cross the y-axis? ________________________________ Why doesn’t the graph of a hyperbola cross the x-axis? ________________________________ As x approaches zero from the left, what happens to the y-values? _______________________ As x approaches zero from the right, what happens to the y-values? ______________________ As x increases in the positive direction, what happens to the y-values? ____________________ As x decreases in the negative direction, what happens to the y-values? ___________________ A hyperbola has no output value at x = ______ because _______ is undefined. Transformations of Hyperbolas Of course the graph of y = 1/x can be transformed just like any other function. In the next lesson you’ll learn more about these transformations. More specifically, you’ll learn how to graph a transformation of y = 1/x , as well as write an equation for a hyperbola given its graph. 82 FST Name: Transformations of Hyperbolas In the previous lesson you were introduced to a new type of function, the rational function. There are many types of rational functions. The simplest is the hyperbola, which is represented by the parent function y = 1/x. In this lesson you are going to find out what happens to the equation of a hyperbola after it is transformed in one of several different ways. Investigation #1: 1. Graph the function y 1 on the graph to the right. x2 2. Compare this graph to the graph of the parent function. What type of transformation occurred? _______________ How does the “2” in the equation you graphed relate to the transformation? _________________________________ What happened to the asymptotes of the parent function? ______________________________________________ Describe what the graph of y 1 looks like. ______________________________________________ x5 Investigation #2: 1. Graph the function y 1 3 on the graph to the right. x 2. Compare this graph to the graph of the parent function. What type of transformation occurred? ______________ How does the “3” in the equation you graphed relate to the transformation? ________________________________ What happened to the asymptotes of the parent function? ______________________________________________ Describe what the graph of y 1 6 looks like. _____________________________________________ x OK. Let’s review. If a number is added to the independent variable what happens to the graph of the hyperbola? ___________________________________________________________________________ If a number is added to the parent function what happens to the graph of the hyperbola? ___________________________________________________________________________ 83 FST Name: Determining the Equation of a Transformed Hyperbola Suppose you are given the graph of a transformed rational function and you must find its equation. The first thing you should do is identify the translations by looking at the locations of the asymptotes (this is what you did above): A horizontal asymptote of y = k indicates a vertical translation of k units A vertical asymptote of x = h indicates a horizontal translation of h units. You might remember that there is another type of transformation other than a translation. It is the stretch. To identify the stretch factor, locate the vertices of an unstretched hyperbola after a translation. You can do this easily because the vertices of the parent function are at (1, 1) and (-1, -1). These points will move the same distance as the translation and will still be 1 horizontal and 1 vertical unit from the center. Then, find a point on the stretched graph that has the same x-coordinate as the translated vertices. The vertical distance from the horizontal asymptote to the point you just found on the graph is the vertical scale factor. An example is shown below. You will find another example on page 539 of your book. The parent function has been translated 3 units up and 3 units left, putting the new “center” at (-3, 3). An unstretched rational function would have vertices at (-4, 2) and (2, 4), 1 horizontal and 1 vertical unit from the center. Because the distance from the point on the graph above (-2, 4) is 2 vertical units from the horizontal asymptote, the vertical stretch is 2. The same case could be made about the point below (-4, 2). The new equation is y 3 2 . x3 Putting it all Together Give the transformations (translations and stretches) for the following hyperbolas. Then, provide the equation of the asymptotes and a “neat” graph. 1. y 1 2 x 1 2. y 3 1 x2 Transformations: Transformations: Asymptotes: Asymptotes: 84 FST Name: Changing Forms of Rational Functions Remember that hyperbolas can always be traced back to the parent function y 1 . x However, some hyperbola equations are written in a form that makes it difficult to see what transformations have been applied to the parent function. This lesson will show you how to re-write these equations so identifying the transformations is easy. Example A: a. Rewrite the function y 3x 11 1 as a transformation of the parent function y . x3 x y 3x 11 x3 Original Equation. We’ll call this rational form. y 3( x 3) 2 x3 Rewrite the numerator by replacing “x” with “x+3”. (You always want to replace the “x” in the numerator with the expression that is in the denominator.) y 3( x 3) 2 x3 x3 y 3 2 x3 Separate the expression into two fractions with the same denominator. Rewrite 3( x 3) as “3” after canceling the “x+3”. x3 We’ll call this transformation form. b. Describe the function using the “transformation” equation. The parent function has been: vertically stretched by a factor of 2 translated left 3 units and up 3 units It has a: vertical asymptote at x = -3 horizontal asymptote at y = 3 Notice in the example above that the equation of the vertical asymptote tells you something about the horizontal translation (left 3 units corresponds to a vertical asymptote of x = -3). There is a similar relationship between the horizontal asymptote and the vertical translation. You’ll have a chance to practice this in a bit, but first we must show you one more thing. 85 FST Name: Finding the x- and y-intercepts of a Transformed Hyperbola To find the x-intercepts of a transformed hyperbola, it’s actually easier to use the rational form. For any x-intercept, the y-coordinate will be zero. To make y = 0, you need to set the numerator equal to zero and solve for x. This x-value is the y-intercept of the hyperbola. To find the y-intercept you can use either form. Set x = 0 and solve for y. Example B: Find the x- and y-intercept(s) of a hyperbola whose equation is: y 3 18 . x3 Hint: To find the x-intercepts, you need to transform this equation to rational form first. To do that, find a common denominator so you can combine both terms into one. Then you can set the numerator equal to 0 and solve for x. OK. We’re finally ready to put everything together from this lesson and those previous. Here we go… Example C: For each problem do the following three things: a. Rewrite the equation in transformation form. Show your work to the right of the problem! b. Describe the graph using the new equation. Be sure to include the direction and distance of any translation, the equations for the asymptotes, the magnitude of any stretches, and the coordinates of any x- and y-intercepts. c. “Neatly” draw the graph and check your description’s accuracy. 1. y 2x 3 x 1 Equation: Description: 86 FST Name: 2. y 5x 14 x2 Equation: Description: 3. y 4 x 4 x2 Equation: Description: 87 FST Name: Graphs of Rational Functions In the last section, we took a close look at one special type of rational function—the hyperbola. We saw a much different type of graph than we are normally accustomed to. This graph had 2 parts, or “branches.” In other words, the graph was discontinuous. This means there was a break in the graph (you couldn’t draw the graph without picking up your pencil). The discontinuity came in the form of the horizontal and vertical asymptotes. In this section we are going to take a look at the graphs of several different types of rational functions. Again, we can expect to see asymptotes, but there are some other features that we’ll discover that are unique to rational functions. One of these features is called a hole. A hole is a single missing point on the graph of a function. Before moving on, explain why graphs of rational functions discontinuous.______________________________________ Investigation—Predicting Asymptotes and Holes In this investigation you will consider the graphs of four rational functions. Step 1: Match each rational function with a graph. Use a friendly window as you graph and trace the equation on your calculator. Describe the unusual occurrences at and near x = 2, and try to explain what feature in the equation makes the graph look the way it does. (You will not actually see the hole pictured in graph d unless you turn off the coordinate axes on your calculator.) 1 1. y ( x 2) 2 a. ( x 2) 2 3. y x2 1 2. y x2 Function: __________________ Description: b. Function: _________________ Description: c. Function: _________________ Description: 88 4. y x2 x2 FST d. Name: Function: _________________ Description: Step 2: Find a rational function for each graph. Write a few sentences that explain the appearance of the graph. a. Function: _________________ Description: b. Function: _________________ Description: c. Function: _________________ Description: d. Function: _________________ Description: Step 3: Write a statement explaining how you can use an equation to predict where asymptote(s) will occur. 89 FST Name: Write a statement explaining how you can use an equation to predict where a hole will occur. Step 4: Graph the following function. y x2 ( x 2) 2 Does it have a hole or an asymptote? _______________________ How can you use an equation to determine if the graph will have a vertical asymptote or a hole? Putting it all Together Complete the flowchart below that describes how to determine the nature of the discontinuity for a rational function. First, determine the degree of each factor in the numerator and the denominator. Then compare the degree of the common factors. Degree of numerator is Degree of denominator is _________________ _____________________ A ________________ A ________________ occurs at the x-value that causes you to occurs at the x-value that causes you to __________________ __________________ 90 FST Rewriting Rational Functions Name: Sometimes you will need to factor the numerator and/or the denominator of a rational function before you can easily determine where holes and asymptotes will occur. This lesson has you practice doing just that. Example 1: Describe the features of the graph of y x2 x 6 . x 2 5x 6 First, factor both the numerator and the denominator: y= There is a hole at ___________ because it is a zero occurring the same number of times in both the numerator and the denominator. There is a vertical asymptote at ____________ because the factor ____________ is in the denominator, but not the numerator. There are several other important features of the graph we can discuss now that the rational function is written in factored form. First is the function’s x-intercept. The function’s x-intercept occurs when y = _______. How could you use the function to help you find the x-intercept? Show your work below. To find an x-intercept look for a factor that is in the numerator but not the denominator. Set any of these factors equal to zero and solve for x. Those are the x-intercepts. The second important feature is the function’s y-intercept. It occurs when x = _______. Show your work in finding the y-intercept below. The last important feature is the horizontal asymptote(s). To find these, consider what happens to the y-values as the x-values get very far from 0. The y-values get close to 1, so there is a horizontal asymptote at y = 1. You can use your calculator’s table feature to help you find this information. 91 FST Name: Now it’s your turn! Change the following functions to factored form. Then give a complete description (x-intercepts, y-intercepts, holes, and asymptotes) of the function. By the way, not every function will have all of these features. a. y x2 x 1 x2 1 x 2 3x 10 b. y x2 c. y x 2 2x 3 x 2 2x 8 92 FST Slant Asymptotes Name: Just like vertical and horizontal asymptotes, slant asymptotes are lines the graph approaches. They are also called oblique asymptotes. A graph has a slant asymptote if the degree of the numerator is bigger than the degree of the denominator AND the factors in the numerator do not “cancel” all of the factors in the denominator. If a graph has a slant asymptote it will not have a horizontal asymptote. To find a slant asymptote, divide the numerator by the denominator and keep only the quotient (the answer). Throw away the remainder. Don’t forget that these are still lines, so they are written as “y = mx + b.” To divide, you either have to use long division or synthetic division (if possible). Example 1: Find any slant asymptotes. Show all work. 3x 3 x2 1 a. y b. 2x2 y x 1 c. y x2 9x 2 x4 93 FST Name: The following function will NOT have a slant asymptote, even though the degree of the numerator is bigger than the degree of the denominator. Show why this is true. x 3 2 x 2 5x 6 d. y x2 What kind of graph will the function above be? _____________ Why? ______________ Are there any discontinuities? ________ Type? ____________ Location? ___________ A Different Type of Rational Function Functions can be written so only part of the function is “rational.” 1 Graph y x and describe the asymptotes. x Next, graph y 2 x 3 2 and describe its asymptotes. x 1 How can you determine the slant asymptote by just looking at the equation? ______________________________________________________________________ How does the “rational part” of the function help you make an accurate graph? ______________________________________________________________________ 94 FST One Last Super Fun Thing! Name: Wouldn’t it be cool if we gave you a graph and asked you to write an equation for it? Example 2: Given the two graphs below (which are of the same function): a. Which graph (L or R) makes it easier to find the slant asymptote? ____________ b. Find the equation of the slant asymptote. __________________ c. Which graph (L or R) makes it easier to find the vertical asymptote? ____________ d. Find the equation of the vertical asymptote. _________________ e. Give the coordinate(s) of the x-intercepts (zeros). ________________ f. Give an example of an equation with asymptote x = -2. ___________________ g. Name a polynomial with zeros at x = -3 and x = 1. _______________________ h. Put all of this together to write an equation for the function shown. ____________________________________ 95 FST Adding and Subtracting Rational Expressions Name: Adding and subtracting rational expression is very similar to adding and subtracting fractions with whole numbers. In particular, you must first rewrite the expressions so that they have a common denominator. Once you’ve done this, you simply add (or subtract) the numerators. Example A: Find the least common denominator of the following expressions. a. 4 and 10 b. (x – 3)(x + 4) and (x – 3) c. (x –4)(x + 5) and x2 – 16 ___________ _______________________ ______________________ Example B: Combine the two rational expressions on the right side of the equation into a single rational expression in factored form. y 7 2x 5 ( x 4)( x 3) x 3 The least common denominator is: ____________________. Example C: Combine the two rational expressions on the right side of the equation into a single rational expression in factored form. y x3 2x 1 ( x 1)( x 2) ( x 2)( x 2) The LCD is: ______________________________. 96 FST Name: So, why are we learning how to do this? I’m glad you asked! We’re learning how to do this because it is much easier to see what the graph of a rational expression looks like if there is only one fraction. Example D: Find any x- and y-intercepts, asymptotes, and/or holes in the graph of: y x2 5 ( x 3)( x 4) x 1 97 FST Multiplying and Dividing Rational Expressions Name: To multiply and divide rational expressions, you do not need to find a common denominator. You simply multiply the numerators and then multiply the denominators. Example A: 5 8 = 7 15 5 3 = 12 4 When you are working with rational expressions, it is best to factor all of the expressions first. This will make it easy to reduce common factors potentially resulting in fewer things to multiply. Example B: x2 7x 6 2x2 2x x 2 5x 6 x 1 Example C: ( x 2) ( x 1)( x 3) ( x 3) ( x 1)( x 2 4) To divide fractions, remember that you simply invert the “lower fraction” and multiply. 3 8 = 9 16 2 Example D: 5 = 6 7 Again, when working with rational expressions, it is best to factor everything first. Example E: x 2 8x 9 x2 x2 2x 3 x 2 7 x 18 Example F: 98 x2 1 x 2 5x 6 x 2 3x 2 x3 FST Name: Once you get rational expressions written as ONE fraction (instead of 2 or more that are multiplied or divided), you can use the skills you learned in the previous lessons to identify the important features of the graph (holes, intercepts, asymptotes, etc.). You’ll get a chance to do this on your assignment, but here is an example: x 2 8x 9 x2 Example G: List the features of the graph of: . (This is the expression from example E.) x2 2x 3 x 2 7 x 18 99 FST Name: Transformations of Hyperbolas Practice 1. Write an equation and graph each transformation of the parent function f(x) = 1 . Use a different color for each graph. x a. Translate the graph up 2 units. ________________________ b. Translate the graph right 3 units. ________________________ c. Vertically stretch the graph by a scale factor of 3. ________________________ 2. Give the equations of the asymptotes for each hyperbola below. a. y 2 1 x b. y 3 x4 c. y 2 4 x3 __________________ ___________________ ___________________ __________________ ___________________ ___________________ 3. As the rational function f(x) = 1 is translated, it’s asymptotes are translated also. Write an equation for the x translation of this function that has the asymptotes given. a. Horizontal asymptote y = 2, and vertical asymptote x = 0 _________________________ b. Horizontal asymptote y = -4, and vertical asymptote x = -3. _________________________ 4. Write a rational equation to describe each graph. Some equations will need scale factors. a. b. _______________ c. ________________ 100 ________________ FST Name: 5. Ohm’s law states that I V . This law can be used to determine the amount of current I, in amps, flowing in the R circuit when a voltage V, in volts, is applied to a resistance R, in ohms. a. If a hairdryer set on high isusing a maximum of 8.33 amps on a 120-volt line, what is the resistance in the heating coils? b. In the United Kingdom, power lines use 240 volts. If a traveler were to plug in a hairdryer, and the resistance in the hairdryer was the same as in 3a, what would the flow of the current be? c. The additional current flowing through the hairdryer would cause a meltdown of the coils and the motor wires. In order to reduce the current flow in 3b back to the value in 3a, how much resistance would be needed? For the next problem, refer to example B on page 540 of your textbook if necessary. 6. The graph at right shows the concentration of acid in a solution as pure acid is added. The solution began as 55 ML of a 38% acid solution. a. How many milliliters of pure acid were in the original solution? b. Write an equation for f(x). c. Find the amount of pure acid that must be added to create a solution that is 64% acid. d. Explain why the solution can never be 100% acid. 101 FST Name: Changing Forms of Hyperbolas Practice For problems 1-4, Rewrite the equation in transformation form. Show your work to the right of the problem. Give a description of the hyperbola. You must include: o The direction and length of any translation(s) o Equations for the asymptotes o Any vertical stretch factor o The coordinates of any x-intercept(s) and y-intercept(s). Show the work needed to find these! Draw the graph to check your description. 1. y 2x 3 x 1 2. y Description: 2x 1 x2 Description: Translations: Translations: Asymptotes: Asymptotes: Vertical stretch factor: Vertical stretch factor: Reflection? x-intercept(s): Yes No Reflection? y-intercepts: x-intercept(s): 102 Yes No y-intercepts: FST 3. y 5x 5 x2 4. y Description: 11 4 x x2 Name: Description: Translations: Translations: Asymptotes: Asymptotes: Vertical stretch factor: Vertical stretch factor: Reflection? x-intercept(s): Yes No Reflection? y-intercepts: x-intercept(s): Yes No y-intercepts: For problems 5 and 6, rewrite the equations in rational form and then find the x-intercept. 5. y 2 6 x4 6. y 5 Rational Form: _________________; x-int: __________ 2x 6 x9 Rational Form: _________________; x-int: __________ 103 FST Name: More Practice Your Skills (Hyperbolas) 1. Write an equation and graph each transformation of the parent function f ( x) 1 . Put graphs on next page. x a. Translate the graph left 4 units. __________________________ b. Translate the graph up 6 units and left 1 unit. __________________________ c. Translate the graph left 3 units and down 4 units. __________________________ d. Vertically stretch the graph by a scale factor of 3, translate it down 2 units and reflected over the horizontal asymptote. __________________________ 2. Write equations for the asymptotes of each hyperbola. a. y 2 x b. y 1 x3 c. y 1 4 x __________________ __________________ __________________ __________________ __________________ __________________ d. y 1 2 x e. y 4 1 x2 f. y 5 2 x4 __________________ __________________ __________________ __________________ __________________ __________________ 3. Rewrite each equation in rational form (top answer blank). Then find the x-intercepts (bottom answer blank). a. y 2 3 x b. y 1 1 x2 c. y 4 2x 7 x5 __________________ __________________ __________________ __________________ __________________ __________________ 104 FST Name: 4. Rewrite each equation in transformation form and then give the vertical stretch, translation(s), and equations of the asymptotes. a. y x x 1 b. y 2x 5 x3 __________________ __________________ __________________ __________________ __________________ __________________ __________________ __________________ 105 FST Quiz Review—Hyperbolas 1. y Name: a a a k is y transformed. is the parent function for a hyperbola when a = 1. y x xh x a. What effect does “a” have on the graph? ___________________________________ b. If a < 0, what change do you see in the graph? ______________________________ c. What effect does “h” have on the graph? When h>0, ______________________________________________________ When h<0, ______________________________________________________ d. What effect does “k” have on the graph? When k>0, ______________________________________________________ When k<0, ______________________________________________________ 2. Write an equation of each transformation of f ( x ) 1 described below. x a. Shift 3 units left, 1 unit down. _________________ b. Stretch by a factor of 4, shift 2 units right. Q2 & Q4 graph. _________________ c. Stretch by a factor of 2, shift ½ unit left, 3 units up. _________________ d. Shift right 3.5 units, up 4 units. Reflected over the horizontal asymptote. _________________ 3. Describe the transformations (stretches and shifts only) given the following equations. Also, indicate whether the graphs have been reflected horizontally. a. y 3 x2 b. y 1 4 ______________________________________________________ x 5 c. y 2 _________________________________________________________ 6 _____________________________________________________ x 8 106 FST Name: 4. For the given functions, name any horizontal or vertical asymptotes. a. y 1 ______________________ x c. y 4 b. y 1 ___________________ x 1 _____________________ x2 d. y 3 2 __________________ x 1 5. Write an equation for a hyperbola with the given asymptotes. a. vertical: x = 3 horiz: y = -1 b. vertical: x = -2 horiz: y = 3 __________________ _________________ 6. Write the function for the following graphs. a. b. _______________ _______________ 7. Change the following functions to rational form (top blank). Then find the x-intercepts (middle blank) and y-intercept(s) (bottom blank). a. y 2 3 x4 b. y 6 2 x 1 c. y 7 3 x 5 _____________ ______________ ______________ _____________ ______________ ______________ _____________ ______________ ______________ 107 FST Name: 8. Change the following functions to translation form. Then give the stretch, shift(s), and the equations of asymptotes for the graph. Also, indicate whether the graph has been reflected or not. a. y 3x 7 x3 b. y x7 x2 New Equation: ___________________ New Equation: ____________________ Stretch: ___________ Stretch: ___________ Shift: ___________________________ Shift: ____________________________ Asymptotes: _____________________ Asymptotes: ______________________ Reflection? Reflection? Yes No 108 Yes No FST Name: Graphs of Rational Functions Practice 1. Without graphing, determine whether the functions below have a hole, a vertical asymptote, both or neither. Explain why you made the choice you did. a. y 2. x3 x3 b. y c. y ( x 1)( x 1) x 1 d. y ( x 2) 2 ( x 2)( x 2) _______________ _________________ __________________ ___________________ _______________ _________________ __________________ ___________________ _______________ _________________ __________________ ___________________ Without graphing the functions above, determine the location of all vertical asymptotes and holes. a.______________ 3. 2( x 1) ( x 1) 2 b.________________ c._________________ d._________________ Write an equation for each graph. a. A horizontal line at y = 1 with a hole at (-2, 1) b. A horizontal line at y = -2 with a hole at (3, -2) c. (Hint: There’s a vertical shift!) ___________________ 4. ______________________ _________________________ This problem is a bit different from those in #1. It should help you understand one additional fact that is very important in helping you determine whether there are holes or asymptotes on a graph. a. For y x2 would you predict a hole or an asymptote? x 1 b. Graph the function. Do you get a hole or an asymptote? __________________ __________________ c. What is different about the numerator and denominator that might explain why the rule we developed in the lesson doesn’t apply here? _____________________________________________________________________________ 5. Factor the following polynomials completely. (These should be a review.) a. x2 + 2x – 3 b. 2x3 – 2x2 – 12x c. 2x2 + 5x – 12 d. 4 – x2 ______________ _______________ ______________ _____________ 109 FST Rewriting Rational Functions Practice Name: 1. Rewrite each rational expression in factored form. a. x 2 7 x 12 = x2 4 b. x 3 5 x 2 14 x = x 2 2x 1 2. Identify the vertical asymptotes for each equation. (These are the equations from problem #1.) a. y x 2 7 x 12 x2 4 b. x 3 5 x 2 14 x x 2 2x 1 3. Graph each equation. Indicate the coordinates of any holes on your graphs. a. y 5 x x5 b. y 3x 6 x2 c. y x 2 x 12 x4 4. Rewrite each function in factored form. Then give a complete description (x-intercepts, y-intercepts, holes, and asymptotes) of the function. x2 x 6 b. y 2 x 2 x 15 x3 a. y 2 x 6x 8 110 FST Slant Asymptotes Practice Name: For the functions in #’s 1-4, find the slant asymptotes without graphing. Show all work. 1. y 3. y x 3 27 x2 3 2x3 7x2 4 ( x 3)( x 1) 2. y x 2 5x 8 x3 4. y x3 x 3 x2 x 2 Answer the following questions. 5. Give all the important features of the following function. y x 4 x3 6. Graph each function. List all holes and asymptotes. a. y x 2 1 x b. y 2 x 3 2 x 1 111 c. y 7 8 4x x2 FST Name: 7. Follow the procedure in the last part of the lesson notes to write an equation for the function shown in the graphs below (both graphs are of the same function). ________________________ 112 FST Rational Functions: More Practice Name: 1. Rewrite each rational expression in factored form. a. d. x 2 5x 6 x 2 25 4 x 3 25x x 2 14 x 48 b. 2 x 2 3x 2 3x 2 5x 2 c. x 2 16 6x 2 7 x 3 e. x 3 5x 2 24 x x 2 6x 9 f. 9x2 1 2 x 3 x 2 3x 2. Rewrite each expression in rational form (as the quotient of two polynomials). a. 2 3 x d. 3x 4 1 2x 3 b. 1 e. 1 x2 c. 4 5x 7 4 x3 2x 7 x 5 f. 6 10 x 3 3x 5 3. Find all vertical and horizontal asymptotes of each function. a. f ( x ) x x 1 b. f ( x ) 2x 5 x3 113 c. f ( x ) 1 x2 FST Name: 3 d. f ( x ) ( x 2) 2 x2 x 1 e. f ( x ) x2 4 f. f ( x ) x3 x 6x 8 2 4. Find all vertical and slant asymptotes of each rational function. a. f ( x ) x2 x 1 d. f ( x ) 2x2 5 x3 b. f ( x ) x2 1 x c. f ( x ) x2 x 1 x 1 e. f ( x ) x3 x2 4 f. f ( x ) 9 x2 2 x 5. Give the coordinates of all holes in the graph of each rational function. a. f ( x ) x3 3 x b. f ( x ) x 5 x 5 x2 4 d. f ( x ) x2 c. f ( x ) 2x 6 x3 x 2 3x 10 e. f ( x ) x2 114 FST Name: 6. For each equation below: i. Rewrite the rational function in transformation form. ii. Describe the graph. Include: translations, asymptotes, stretches, holes, and x- and y-intercept(s). a. y 6 x 13 x2 b. y 115 5x 18 x4 FST Rational Functions Quiz Review Name: 1. For the given functions, name any horizontal, vertical, or slant asymptotes. a. y ( x 1)( x 3) ( x 2)( x 4) ________________________________________________ x 2 3x 6 b. y x2 ________________________________________________ c. y x3 4x2 x 7 x2 1 ________________________________________________ d. y 2x3 3 x2 x 1 ________________________________________________ 2. Write an equation with the given asymptotes. a. vertical: x = 2 slant: y = x + 2 _________________ b. vertical: x = 0 slant: y = 2x + 3 __________________ 3. Which equation has a hole and which has an asymptote? Explain your choices. a. y x 1 ______________ ( x 1) 2 _______________________________________ _______________________________________ b. y ( x 1) 2 ______________ x 1 _______________________________________ _______________________________________ 4. Write the function for the following graphs. a. b. ________________ ________________ 116 FST Name: c. d. _______________ __________________ 5. Give the coordinates of the holes for the following functions. a. y 2x 8 ______________ x4 b. y 3x 3 ______________ x 1 6. Write the rational functions in factored form. Describe the transformations, asymptotes, holes, and x- and y-intercepts. a. y x 2 x 20 x 2 7 x 12 Equation: Description: b. y x2 2x 3 2 x 2 6x 8 Equation: Description: c. y 4 x 3 4 x 2 3x 2x2 9x 5 Equation: Description: 117 FST Adding and Subtracting Rational Expressions Name: 1. Factor each expression completely and reduce any common factors. a. x2 2x x2 4 b. x 2 5x 4 x2 1 c. 3x 2 6 x x 2 6x 8 d. x 2 3x 10 x 2 25 2. What is the least common denominator for each pair of rational expressions? a. x x 1 and ( x 3)( x 2) ( x 3)( x 2) c. 2 x and x 4 ( x 3)( x 2) 2 b. d. x x2 and ( x 1)( x 2) (2 x 1)( x 4) x2 x 1 and 2 x 5x 6 ( x 3)( x 2) 3. Add or subtract as indicated. a. x x 1 + ( x 3)( x 2) ( x 3)( x 2) b. 118 2 x x 4 ( x 3)( x 2) 2 FST c. x2 x 1 + 2 x 5x 6 ( x 3)( x 2) 4. Graph y Name: d. 2x 3 2 ( x 1)( x 2) x 1 x 1 x x 7 x 8 2( x 8) 2 a. List all: -asymptotes -holes -intercepts b. Rewrite the right side of the equation as a single rational expression. c. Use your answer from 6b to verify your observation in 6a. Explain. 119 FST Multiplying/Dividing Rational Functions Practice Name: 1. Multiply or divide as indicated. a. x 1 x2 4 2 ( x 2)( x 3) x x 2 c. x2 7x 6 2x2 2x x 2 5x 6 x 1 d. x3 x2 9 x 2 8 x 15 x 2 4 x 5 e. 2x2 4x 2 x 10 2 2 x 25 3x 5x 2 f. 4x2 x2 4x 4 x2 2 x 3 8x b. x 2 16 x 2 8 x 16 2 x 5 x 3x 10 2. Rewrite as a single rational expression. x x2 x 1 x2 4 1 a. 1 b. x 1 x x 1 120 1 x 1 x x 1 FST Operations with Rational Functions More Practice Name: a. Add, subtract, multiply or divide as indicated. SHOW ALL WORK. b. Then identify any features of the graph that you can. Include asymptotes, holes, x-intercepts, and y-intercept. 1. y = 4 x 2 x 49 ( x 7)( x 1) 2. y = x4 x 2 3x 10 x 2 3x 2 x 2 16 3. y = x2 2x 2 2 x 5x 6 x 9 4. y = x 2 2 x 15 4x2 1 2 x 2 9 x 5 2 x 2 5x 3 NOT DONE YET 121 FST Name: 9 x 6x 6x x 2 2 2x 1 4x 4x 1 2 5. y = 2 6. y = x9 x9 2 2 x 9x x 9x x2 9 2 8. y = x 2 2 x 3 x 6x 9 x 1 2x 1 x2 7. y = 2 2 x 3x 2 x 122
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