Pre Calculus
WorksheetExtra
P.1 Practice
Intervals:
1. Describe in words and graph each interval of real numbers.
a) x < 2
b) –2 < x < 5
Also, in set-builder notation and interval notation
c)
,7
d) [–3, 3]
2. Graph the interval described in words and use interval notation to describe the interval.
a) x is negative
b) x is greater than or equal to 2 and less than or equal to 6
3. Explain why each of the following should NEVER be written:
a) 5 < x > 7
b) 3 > x > 8
4. Describe the graphs below using both inequalities and interval notation.
3
6
9
12
15
18
5
10
15
20
25
30
Later in the course we will be using sign charts. Sign charts are simply number lines with “pos” or “neg” signs on them to represent
whether or not the function or equation is positive or negative on that interval. Suppose the sign chart below represents when the
function f (x) is positive “pos” or negative “neg”. Use the sign chart to answer the questions 5 and 6 in interval notation.
5. When is f (x) < 0 ? … (i.e. when is the function negative?)
f (x)
pos
+
pos
neg
–2
1
pos
7
6. If f (x) = 0 when x = –2 and x = 1, but not when x = 7, when is f (x) > 0 ?
7. Use both inequality AND interval notation to describe the set of numbers. Define any variables used.
a) In Excel, you can create an if-then statement that will enter a specific value or text into a box depending on given
conditions. Suppose you would like to enter a value of “YES” into cell A8 if the value in cell A7 is between 4 and 16
including both.
b) A programmer wants to create an alertbox if the user inputs a value that is at least 150.
8. Identify the base in each of the following expressions, then evaluate each expression.
a) −42
b)
( −4 )
2
c)
( −3)
3
d) −33
e) 2 x + 12
f)
( 2 x + 1)
2
9. Match the following equations with the property illustrated. Each letter is only used once.
3x y
a 2b
3 xy
ba 2
a 2b
x
A) Commutative Property of Addition
3
B) Commutative Property of Multiplication
a 2b
0
0
x
2
C) Associative Property of Addition
3
D) Associative Property of Multiplication
2
E) Identity Property of Addition
a x
y
ay
1
2
x
ax
x
F) Identity Property of Multiplication
G) Inverse Property of Addition
1
2
H) Inverse Property of Multiplication
1 x
a
y
b
x2 y
x
c
z
y
I) Distributive Property
c
a
a3
x2 y
b
z
a3
10. Simplify each expression.
a)
( 4 xy + 2 x ) ax
3
b) 3a ( 2a + 5b )
2
Many times we need to undo the distributive property. This is the beginning of factoring. When you undo the distributive property,
we say you have factored out the greatest common factor.
11. Factor out the GCF from each expression.
a) −3 x 2 − 6 xy
b) 108 x 3 − 36 x 2 + 60 x
12. Describe in your own words what the phrase “Take out a GCF” means.
c)
2
3
6
+
5
2
Pre Calculus
Worksheet P.3
1. Solve each of the following equations algebraically:
a) 4 y
2
2x 3
4
5
d)
5y
b)
1
2
x
3x
7
8
c)
e)
t 5
8
t 2
2
1
2
x
1
3
1
1
3
2. The formula for the perimeter of a rectangle is given by P = 2L + 2W. Solve this equation for W.
3. The formula for the area of a trapezoid is given by A
4. A formula relating Fahrenheit to Celsius is C
5
9
1
2
h ( b1 + b2 ) . Solve this equation for b1.
( F − 32 ) .
Solve this equation for F.
5. A satellite orbiting the planet has a mass m, a velocity v, and a radius from the center of the planet equal to R. The
mv 2
. Solve this equation for R.
centrifugal Force F acting on that satellite is given by the equation F
R
6. You have scores of 68, 82, 87, and 89 on your first four tests, and you have one more to take. In order to get a B you
must have an average between 79.5 and 89.5. What is the lowest score you could get on your last test in order to get a B.
7. When you solve an inequality, when does the inequality change direction. Explain why.
8. Solve each of the following inequalities and write your answer in interval notation.
a) 2
c)
5x 7
4
e)
3 4y
6
x
6
9
3
2y 3
8
1 3x
b)
2y 5
3
d) 4
2
y
f)
1
2
x
3
2
7
2
2 x
4
1
3
x
3
Pre Calculus
Worksheet Factoring
1. Learning to recognize the difference of two perfect squares is more difficult than factoring them. Factor the following
expressions, and write something on this paper that will help you recognize this in the future.
a) a 2 − 25
b)
16 − n 2
2. Factor the following quadratic expressions completely:
a) b 2 + 8b + 15
d)
b)
p3 − 4 p 2 − 5 p
6d 2 + 35d − 6
c) 10t 2 − 19t + 6
e) 12 f 3 − 14 f 2 − 40 f
f) 20e 4 − 23e3 + 6e 2
3. Factor the following equations completely using the grouping method:
a)
b) mp 2 + 7 m + 3 p 2 + 21
ax + ay + 6 x + 6 y
c) b3
7b 2
9b
63
4. Factor each of the following expressions:
a) 6 5 x
3
2
11 5 x
3
7
b) 5 8 x
9
2
47 8 x
9
18
Pre Calculus
Worksheet P.5
The main purpose of section P.5 is for you to be able to solve quadratic functions in the most efficient (quickest)
method possible. In order to do this you should learn as many different methods as possible.
1. Graphically solve the equation x3 + x 2 + 2 x − 3 0 . Describe your method.
2. Graphically solve the equation x 2
3. Solve the equation 2 x
answer(s).
4
4 x . Describe your method.
x 2 graphically two different ways … keep in mind you should get the same
a) Solve by finding the intersections.
b) Solve by finding the zeros.
4. The table below shows the x- and y-coordinates of the equation y x 2 2 x 1 .
State the expression and give the zero as accurately as can be read from the table.
x
0.4
0.41
0.42
0.43
0.44
0.45
0.46
5. Solve each of the following equations by using factoring:
a) x 3 x
11
20
b) x 3
x2
9x
9
0
y
-0.04
-0.0119
0.0164
0.0449
0.0736
0.1025
0.1316
6. Solve each of the following equations by “extracting square roots”.
a) 3 x 2
5
b) 2 x
43
3 2
4
49
32
7. Solve each of the following equations by “completing the square”
a) x 2
6x
b) 3 x 2
7
8. Solve the equation using the quadratic formula: x x
5
x
2
0
12
Keep in mind that for the purposes of quizzes and tests, there will be times that you will be asked to use a
specific method. However, in general, you may use whatever method works for you and is appropriate.
The second portion of this section is for you to solve equations involving absolute value.
9. Solve each of the following equations graphically AND algebraically:
a) t
8
b) 5
2
2 3x
1
c) x 2
9
4x 1
4
10. The following equations represent things you should be able to do to this point. Solve each algebraically:
a)
2 5 3x
d) 3x 1
5
8
4
5x
b)
3 5x2
e) x 2 x
c)
17
3 x
4
3x
4
f) 4 x 2
x
3
6x
1
12
Pre Calculus
Worksheet P.6
For questions 1 – 4, solve the inequality algebraically. Write your answer in interval notation.
1. x 8
17
2. 3 2 x
3. 3 2 x 9
9
4. 5 2 x 3
4 , but the equation x 2 16 has two solutions, x
Using this fact, solve the following inequalities algebraically:
5. The 16
a) x2
9
0
4
8 13
4 . The reason for this is the fact that
b) 2 x 3
2
5
6. Solve the following inequalities graphically. Write your answer in interval notation.
2
a) 2 x
17 x 21 0
c) 2 5x 3x2
8
b) 2 x3
3
d) x
2x
x
0
5
13
x2
x.
7. For a certain gas, P = 400/V, where P is the pressure and V is the volume. If 20 < V < 40, what is the corresponding
range of P ?
8. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 20- by 25-inch sheet
of tin and bending up the sides (see figure below).
x
a) Write an equation for the volume V as a function of x.
What is the domain of this function?
x
x
x
20
"
x
x
x
3
b) What size corner squares should be cut to yield a box with a volume of 300 in ?
x
25
"
c) What size corner squares should be cut to yield a box with a volume more than 300 in 3?
d) What size corner squares should be cut to yield a box with a volume of at most 300 in 3?
e) How large should the square be to make the box hold as much as possible? What is the resulting volume?
Pre Calculus
Worksheet 2.5
1. What is
1?
Just because we refer to solutions as imaginary does not mean that the solutions are meaningless. Fields such as quantum
mechanics and electromagnetism depend on the mathematics of imaginary numbers. When engineers design airplane
wings or cell-phone towers, imaginary numbers are vital to their calculations. Applications abound in electrical
engineering, vibration engineering, polymer science, navigation, and more. Applications abound in the real world,
touching our lives via design of popular features such as the vibrating ringer in our cell phones or the bass boosters in our
MP3 players. More heavy duty applications include the design of missile guidance systems. While we are not ready to
dive into these topics yet, we are ready to learn to simplify the square root of negative numbers correctly.
2. Evaluate the following expressions: (no calculator)
25
a)
60
b)
c)
52
4
6
3
When adding, subtracting and multiplying complex numbers, pay close attention to whether you are adding, subtracting,
or multiplying. MOST people that miss these questions do so because they decide to multiply everything.
4. What is i 2 ?
5. Perform the indicated operation and simplify each of the following expressions:
a) 7
3i
6 i
b)
7
3i
c)
6 i
7
3i 6 i
6. Find the conjugate of each complex number then multiply them together. What do you notice about your answers?
a) 4
9i
b)
5
3i
8. Divide the following complex numbers. Write your answer in the form a + bi.
a)
2 8i
5 3i
b)
2 3i
9 4i
9. Solve the following quadratic equation by using the quadratic formula. Write your answers in a + bi form where
applicable.
x2
x
11
5x
8
Solving quadratic equations can also be done by using completing the square. Completing the square is most useful when
the coefficient of x2 is equal to 1 (and the coefficient of x is even).
… “Go back to Section P5 if you need a refresher on how to complete the square.”
10. Solve the following equation by completing the square and compare it to question 9a above:
x2
x
11
5x
8
12. Draw a picture of (or explain why you are not able to draw) each of the following:
a) a quadratic function having only one real number root.
b) a quadratic function having only one complex root.
c) a quadratic function with two real roots.
d) a quadratic function with two complex roots.
13. Previously, we used synthetic division to show that a value of x was indeed a zero of a function. If you were told that
one of the solutions to the equation x 2 4 x 5 0 is x = 2 + i, use synthetic division to show that it really is a zero.
Using the result of your synthetic division, what is the other solution?
: You will use
synthetic division with
complex numbers in
lesson 2.6
In the last example the conjugates x = 2 + i and x = 2 – i were both solutions. This is not a coincidence. In a polynomial
if x = a + bi is a zero, then the conjugate x = a – bi is also a zero. Using the last example, this means that the expression
x 2 4 x 5 could be factored to x 2 i x 2 i .
14. Write the polynomial function of minimum degree in standard form whose given zeros and their multiplicities include
those listed below.
a) x = 4i
b) x = –5 and x = 4i
c) x = –2 (multiplicity 2), and x = 2 + 3i
15. Given the complex numbers w = – 2 + 4i and v = 5– 2i, answer the questions below…
a) Plots points in the complex plane.
Imaginary axis
Real axis
b) Find 5 − 2i and explain what this means.
c) Find the distance between w and v.
d) Find the midpoint of the segment connecting w and v.
OPTIONAL EXTRA PRACTICE:
For more practice on each of the following topics…refer to pages 234-235 of your book.
Given zeros of a polynomial, find the equation of the polynomial… #5–15
Pre Calculus
Worksheet 2.6
1. State the Fundamental Theorem of Algebra in your own words. Why do we need it?
2. Use the equations and the graphs below to do the following:
i) Identify the number of complex zeros each equation has according to the Fundamental Theorem of Algebra.
ii) Determine how many real and non-real zeros each function has according to the graph.
a) f ( x )
x − 2x − 7x + 6
4
2
b) f ( x )
x + 3x − x + 3
3
2
3. How many real zeros are you guaranteed to have if you have an odd degree polynomial?
4. Is it possible to find a polynomial f (x) that has the degree 4 with the given zeros? Explain why or why not.
a) –3, 1 + 2i and 1 – i
b) 1 – 3i and 4 + i
5. For question 4, if there was a possible polynomial with the given zeros, find that polynomial in standard form.
[No Calculator] Using the given zero, find all complex zeros of the polynomial function.
4
3
2
6. f ( x ) 5 x + 3 x + 3 x + 3 x − 2 ; zeros: x = –1 and
8. Use factoring to find all the zeros of f ( x )
9. Use the graph below to help factor g ( x)
2
5
7. h( x)
x 4 − 4 x3 − 4 x 2 + 36 x − 45 ; zero: x = 2 + i
x3 + 3 x 2 + 4 x + 12
3 x 4 − 11x 3 + 10 x 2 − 4 x + 8 . The scale on the x-axis is 1 unit / tick.
Use your graphing calculator to find and verify all zeros. Then, write the function as a product of linear and irreducible
quadratic factors. IF YOU DON’T KNOW WHAT “LINEAR AND IRREDUCIBLE QUADRATIC FACTORS”
MEANS, please ask … yes, now!!!!!!
10. f ( x)
x3
x2
x 15
11. f ( x )
x + 3x − 3x + 3x − 4
4
3
2
12. Find the unique polynomial function of degree 4 with zeros x = 3, x = –1, and x = 2 – i where f(0) = 30. Write your
function as both the product of linear and irreducible factors, as well as, in standard form.
OPTIONAL EXTRA PRACTICE:
For more practice on each of the following topics…refer to pages 234-235 of your book. (If you need one, just ask)
Find zeros of a polynomial given complex zero…#27–32
Factor a polynomial into linear factors and/or irreducible quadratic factors…#37-40
Pre Calculus
Worksheet 2.8
1.
3.
5.
7.
5
6
2m
2m 3
2
19
6
4
8
x 3
x
3
2x
x 3
6
x
x2
x
2
9
6.
3x
1
5
4.
18
3x
x
2.
x
2
x
2
7
3 x 10
8.
4
c
c
3
2
3
3
x
x
4
x
3
x
10
2y 8
2x
x
2
3
x2
x
7y 8
y 2 16
2
x 12
3
x 1
8
2y 8
9.
1
x( x 4)
x2
10. Solve for x: y
5 2x
3x 4
1
5
x( x 1)
1
1
x
11. The total electrical resistance R of two resistors connected in parallel with resistances R1 and R2 is given by
1
R
1
R1
1
R2
One of the resistors has a resistance of 2.3 ohms. Let x be the resistance of the second resistor.
a) Express the total resistance R as a function of x.
b) Find the resistance of the second resistor if the total resistance of the pair is 1.7 ohms.
12. Consider all the rectangles with an area of 200 m2. Let x be the length of one side of such a rectangle.
a) Express the perimeter P as a function of x.
b) Find the dimensions of a rectangle whose perimeter is 70 m.
c) Find the dimensions of the rectangle that has the least perimeter. What is the least perimeter?
13. Explain the difference between the following three problems. Are there any similarities in their solution?
a) Write the expression as a single quotient:
b) Solve:
c) Solve:
x
x
x
2
x
2
x
x
4
x
2
x
2
4
0
0
x
x
x
2
x2
4
Precalculus
Worksheet 2.9
1. Use the graph of g (x) at the right to answer each question.
a) When is g (x) > 0 ?
b) When is g (x) > 0 ?
c) When is g (x) < 0 ?
d) When is g (x) < 0 ?
x
2. Solve the following inequality algebraically:
3
2
x 1 x 1
0 .
3. Compare your answer from question 2 to question 1c. What do you notice?
4. Solve the following inequality using a graph AND algebraically: 5
5. Consider the function f x
2x x
x
4
7 3x
5
x 2x
.
a) When does f (x) = 0 ?
b) When is f (x) undefined?
c) Create a sign chart to determine when f (x) > 0 .
d) Use the sign chart you created in part c to determine when f (x) < 0.
3
3
0
6. Solve each inequality by first completing factoring the left side and creating a sign chart.
a) x 3
4x2
x
4
0
b)
x2
x2
3 x 10
6x 9
0
7. A sign chart in question 5b involved a rational function that was less than zero. You may first need to add or subtract
fractions in order to create a rational function. You may also need to make one side equal 0 before using a sign chart to
solve. Solve the following problems using a sign chart.
a)
1
x
2
2
x 1
0
b)
4x 5
x 2
3
Need more practice? Ask your teacher to check out a textboo if you don’t have one...
Like #2 and #4: Already Factored Polynomial Inequalities: page 265 #7 and 8
Like #6a: Factoring Required Polynomial Inequalities: page 265 #9 – 20
Like #5 and #6b: Rational Inequalities (with or without factoring): page 266 #25, 26 – 40
Like #7: Write the Rational Inequality as a Single Fraction First: page 266 #47, 49
Pre Calculus
Worksheet 3.5
Use the properties of logarithms to expand the expression. Simplify, if possible.
(
1. log 3 81x 5
3
y6
)
2. ln
6 x1/3
eh 5
In 3-4, condense the expression to write the logarithm of a single expression. Simplify, if possible.
3. ln18 − 2 ln 6 + ln 4
For 5-6, let x
5. log b
log b 3 and y
9
4
4. log b 5 − 3 log b x − log b y
log b 4 . Expand the logarithm. Then, write each expression in terms of x and y.
6. log b 0.75
7. Evaluate each logarithm with your calculator. Explain why your answer is reasonable. No, the reason is not
because the calculator said so!
a) log 3 29
b) log 0.4 5.2
There are two algebraic ways to solve an exponential equation…
8. Apply a common logarithm to both sides of the exponential equation: 1.05 x
solve for x.
2 . Use the properties of logarithms to
9. Put the equation 1.05 x 2 into logarithmic form. Solve for x with your calculator. What do you notice about your
answers to questions 8 and 9?
There are two algebraic ways to solve a logarithmic equation…
10. Apply a base to both sides of the logarithmic equation to “undo” the logarithm: ln x
3.5 . Solve for x.
11. Put the equation ln x 3.5 into exponential form. Explain how this process is the same as that of the previous
question and how it is different. What do you notice about your answers to questions 1 and 2?
Solve each equation algebraically showing all work. Round final answers to the thousandths. Remember to check
your answers!
4x
x
12. 8
20(0.5)
13. 9 5
4
15. log 5 (1 − 2 x )
x +1
16. 2 ln
3
18. log ( 5 x ) + log ( x − 1)
2
( 7x )
14. −1 + 6e 3
72
17. 4 − 9 log 7 ( 5 x − 3 ) 1
6
19. log 3 ( a + 2) − log 3 ( a − 3)
2
20. log 4 ( − x )
2 − log 4 ( x + 10)
Examine the following questions, noting how they are different from questions 18-20. Then, solve.
21. log 3 5 − log 3 x
log 3 2
43
22. 3 log 5 2 + log 5 x
log 5 (24 + 2 x )