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Algebra II Chapter 5 5.1 Quadratic Functions The graph of a quadratic function is a parabola, as shown at rig. Standard Form: f ( x) = ax 2 + bx + c ⎛ b ⎛ −b ⎞ ⎞ vertex: (x, y) = ⎜ − , f ⎜ ⎟ ⎟ ⎝ 2a ⎝ 2a ⎠ ⎠ a<0 graph opens down a>0 graph opens up b axis of symmetry: x = − 2a Larson Text, Ch 5, p. 249 Graph each function given in Standard Form: 1. y = x 2 + 4x + 3 2. y = x 2 + 6x + 11 a= a= b= b= x=− b = 2a ⎛ b⎞ y = f ⎜− ⎟ = ⎝ 2a ⎠ b = 2a ⎛ b⎞ y = f ⎜− ⎟ = ⎝ 2a ⎠ (x, y) = (x, y) = x=− Vertex form for a quadratic function (Let’s think about absolute value functions again first to help us understand this.) Recall: Vertex form for the equation of an absolute value function is f (x) = a x − h + k What do the constants in the equation represent? Graph the following absolute value equations using the vertex form. 1. f (x) = −2 x − 1 + 3 2. f (x) = 1 x +1 − 5 3 vertex = vertex = slope of the right branch of the graph = slope of the right branch of the graph = Vertex Form: f ( x) = a( x − h) 2 + k vertex: (h, k) axis of symmetry: x = h a<0 graph opens down a>0 graph opens up Graph each function given in Vertex Form. 1. y = −(x − 1)2 + 4 2. (h, k) = (h, k) = (x, y) = (x, y) = a= a= 1 y = (x + 3)2 − 2 2 Intercept Form: f (x) = a(x − r1 )(x − r2 ) x-intercepts: (r1 , 0) and (r2 , 0) ⎛r +r ⎛ r + r ⎞⎞ vertex: ⎜ 1 2 , f ⎜ 1 2 ⎟ ⎟ ⎝ 2 ⎠⎠ ⎝ 2 a<0 a>0 graph opens down graph opens up Graph each function in Intercept Form. 1. f (x) = (x − 2)(x + 4) 2. f (x) = −2(x + 1)(x − 5) r1 = r1 = r2 = r2 = vertex = vertex = a= a= 5.2 Solving Quadratic Equations by Factoring Zero Product Property Let A and B be real numbers or algebraic expressions. If AB = 0, then A = 0 or B = 0. Solve each equation by using the Zero Product Property: 1. 0 = (x − 1)(x − 3) 2. 0 = x(x + 4) 3. 0 = x2 − 9 4. 0 = x 2 + 3x − 18 5. 2x 2 − 17x + 45 = 3x − 5 6. 4x 2 + 12x − 7 = 0 7. 0 = 6x 2 − 16x + 8 8. 125x 2 − 5 = 0 5.3 Solving Quadratic Equations by Finding Square Roots Simplify expressions using properties of square roots: 1) 48 2) 90 3) 3 16 4) 25 3 5) 6 i 10 6) 5 3 125 Solve the quadratic equation by finding square roots. 7) 2x 2 + 1 = 17 8) x 2 − 9 = 16 9) 4x 2 + 7 = 23 10) 5(x − 1)2 = 50 11) 1 (x + 8)2 = 14 2 5.5 Completing the Square Factor the Perfect Square Trinomials. A.) x 2 − 8 x + 16 = B.) x 2 + 5 x + 25 = 4 C.) x 2 − 7 x + 49 = 4 Can you see any pattern on how the second term in factored form is related to the middle term of the original quadratic? Find the value of c that makes the quadratic equation a perfect square trinomial. Then write the quadratic in vertex form. 1.) y = x 2 − 14 x + c 2.) y = x 2 + 10 x+c 3 Completing the Square to Graph a Quadratic Function Rewrite the equation in vertex form by completing the square. Find the vertex. Then solve for the x intercepts. Verify your x intercepts on the calculator. Graph the parabola. 1.) y = x 2 + 10 x − 3 Vertex Form :______________ (h, k): a= (x, y): 2.) y = x 2 + 6 x − 8 Vertex Form :______________ (h, k): (x, y): a= 3.) y = −x 2 + 4x − 1 Vertex Form :______________ (h, k): a= (x, y): 4.) y = 2 x 2 − 12 x + 14 Vertex Form :______________ (h, k): (x, y): a= 5.) y = 4x 2 − 6x + 1 Vertex Form :______________ (h, k): a= (x, y): 6.) y = −3x 2 − 6x − 8 Vertex Form :______________ (h, k): a= (x, y): Completing the Square to solve a Quadratic Equation Solve the following equations by completing the square. 1.) x 2 − 12x = −28 2.) x 2 + 3x − 1 = 0 3.) −3x 2 + 24x = 27 4.) 4x 2 − 40x − 8 = 0 5.) 3x 2 − 26x + 2 = 5x 2 + 1 6.) 2x 2 + 3x + 1 = 0 7.) −4x 2 − 2x = −5 8.) −3x 2 + 5x = −7 5.4 Complex Numbers (part 1) What happens when you try to solve: x 2 = −1 ??? By definition: i = −1 i = 2 ( ) 2 −1 = −1 Simplify the following radicals. Give your answer in terms of i. −25 −12 −48 From Larson Textbook, page 272 Solving Equations over the Complex Numbers Solve the equation for x, giving your answer in terms of i. 1.) x 2 + 64 = 0 Check: 2.) 2x 2 + 26 = −10 Check: 3.) 3x 2 + 10x = −26 4.) 6x 2 − 2x + 2 = 4x 2 + x 1 5.) − (x + 1)2 = 5 2 6.) −6(x + 5)2 = 120 Plot the complex numbers in the complex plane: imaginary axis 1. 2 – 3i 2. -3 + 4i 3. 2i 4. -i + 7 real axis Adding and subtracting complex numbers: Combine the real parts and combine the imaginary parts. 1. (3 + 4i) + (6 + i) 2. (1− i) − (7 + 3i) 3. (6 − 2i) − (5 + i) − (10 + 5i) 5.4 Complex Numbers (Part 2) Multiplying Complex Numbers Simplify each expression as a complex number in standard form: a + bi 1. 3i(5 − i) 2. −7i(3 − 2i) 3. (2 + 3i)(5 − 6i) 4. (2 − 5i)(2 + 5i) 5. (1+ i)(1− i) Write each expression as a complex number in standard form: a + bi 1. 3 1+ i 2. 6+i 1− 2i 3. 5+i −3 − i 4. 8−i 8+i 5. 6+i 2i 6. 6 i Why do we need complex numbers? Complex numbers are at the heart of understanding Fractal Geometry. See pages 275 and 276 of your textbook. Fractal Geometry is used to model a variety of natural phenomena. Check out this video: http://www.youtube.com/watch?v=ZbK92bRW2lQ Complex numbers are also used in electronics to describe electrical circuits. Complex numbers are used in a variety of higher level mathematics. 5.6 The Quadratic Formula Find the x-intercepts for the following equations by completing the square : y = −2 x 2 − 2 x + 7 y = ax 2 + bx + c Quadratic Formula: y = ax 2 + bx + c −b ± b 2 − 4ac x= 2a Use the quadratic formula to solve for x in each equation. 1. x 2 + 3x − 2 = 0 2. −7x 2 + 2x + 9 = 0 a = _____ b = _____ c = _____ a = _____ b = _____ c = _____ x= x= 3. 5x 2 + 9x = −x 2 + 5x + 1 a = _____ b = _____ c = _____ x= Use the quadratic formula, factoring or taking square roots to solve for x in each equation. Use your graphing calculator to check your solutions. (Graph the equation using y = , then use 2nd Calc, 2: Zero to find where the graph crosses the x-axis.) 1. 5(x − 2)2 + 1 = 0 3. 2x 2 + 5x + 1 = 0 2. 2x 2 + 5x − 3 = 0 5.6 The Discriminant For each equation, find the value of the discriminant, determine how many solutions, and then find the solutions. Check your answers using your graphing calculator. 1. x 2 − 6x + 10 = 0 2. x 2 − 6x + 9 = 0 3. x 2 − 6x + 8 = 0 5.6 Motion Problems 5.7 Graphing Quadratic Inequalities Warm up: Graph the following parabola by finding the vertex and the x-intercepts: 1.) y = x 2 − 2 x − 3 Vertex: _______________ x-intercepts: ____________ Now what if the original equation was written in the form: y ≥ x 2 − 2x − 3 How would that change the graph? Choose 2 points inside the parabola: Choose 2 points outside the parabola: 1.) 1.) Solution? _________ Solution? _________ 2.) 2.) Solution? _________ Solution? _________ Graph the following quadratic inequality or system of quadratic inequalities: 1.) y < 2 x 2 − 5 x − 3 Vertex: __________ X intercepts: _______ 2.) A.) y ≥ x 2 − 4 B.) y < − x 2 − x + 2 Equation A Equation B Vertex: _____ Vertex: _____ x ints: _____ x ints: ______ 3.) A.) y ≤ − x 2 + 9 B.) y > x 2 + 5 x − 6 Equation A Equation B Vertex: _____ Vertex: _____ x ints: _____ x ints: ______ 5.7 Solving Quadratic Inequalities You can easily solve quadratic inequalities by using the following algebraic method. Step 1: Step 2: Step 3: Step 4. Step 5: Factor the quadratic. Find the zeros of the quadratic and use them as boundaries for your solutions. Plot the boundaries on a number line. Use a test point to test the intervals between the boundaries. Summarize your solutions as a compound inequality or using set notation. FYI: What is set notation?? 3≤ x ≤ 5 −1 ≤ x ≤ 10 4<x<7 −1 ≤ x < 8 5<x≤6 can be written as [3,5] can be written as [-1, 10] can be written as (4, 7) can be written as [-1, 8) can be written as (5, 6] Use a bracket [ ] for less than or equal to, use a parenthesis ( ) for less than. Easy! 1. 2x 2 − 7x + 3 ≥ 0 2. 3x 2 − 16x + 5 ≤ 0 3. −x 2 − 12x < 32 −x 2 + x + 5 < 0 4. (If you can’t easily factor, use the quadratic formula to find the zeros.) J 5.8 Modeling with Quadratic Functions WRITING EQUATIONS OF QUADRATIC EQUATION 1.) Write an equation in vertex form for the parabola shown. Recall vertex form: y = a(x − h)2 + k 2.) Write the equation of the parabola in vertex form with vertex (-2,3) and passes through the point (2,-5). 3.) Write an equation in standard form for the parabola shown. Recall standard form: ax 2 + bx + c = 0 4.) Write the standard form of the equation of the parabola with xintercepts of (-3,0) and (2,0) and passes through the point (-1,4). 5.) Write the intercept form of the equation of the parabola with x1 1 intercepts of ( ,0) and ( − ,0) and passes through the point (-1,2). 3 5 Recall intercept form: y = a(x − r1 )(x − r2 ) 6.) A study compared the speed x (in miles per hour) and the average fuel economy y ( in miles per gallon) for cars. The results are shown in the table. Speed,x Fuel economy,y 15 22.3 20 25.5 25 27.5 30 29 35 28.8 40 30 45 29.9 50 30.2 55 30.4 60 28.8 65 27.4 70 25.3 a.) Graph a scatter plot of Fuel Economy vs. Speed on your graphing calculator. Do you think that a linear or quadratic regression model fits the data better? b.) Find a linear regression model for the data using your graphing calculator. Report the equation of the linear model below. Report the r2 value for this model. c.) Find a quadratic regression model for the data using your graphing calculator. Report the equation of the quadratic model below. Report the r2 value for this model. d.) Using the model that fits the data best, predict the fuel economy for a car travelling at a speed of 80 mph. e.) Find the speed that maximizes fuel economy.