Pre-Calculus Summer Assignment
Pre-Calculus Summer Assignment Page 9
State each relation as a set of ordered pairs. Then state the domain and range.
6.
X
-3
0
3
6
Y
4
0
-4
-8
Ordered pairs:
Ordered pairs:
Domain:
Domain:
Range:
Range:
State the domain and range of each relation. Then state whether the relation is a function. Write yes or no.
Explain.
10. {(1, 2), (2, 4), (-3, -6), (0,0)}
11. {(6, -2), (3, 4), (6, -6), (-3, 0)}
10. Domain:
11. Domain:
Range:
Range:
Function: Yes or No
Function: Yes or No
Explain:
Explain:
Evaluate each function for the given value.
13. f(-3) if f(x) = 4x3 + x2 – 5x
15. State the domain of f(x) =
14. g(m + 1) if g(x) = 2x2 – 4x + 2
x 1.
1
Pre-Calculus Summer Assignment Page 9
For each graph, state the domain and range of the relation. Then explain whether the graph represents a
function.
Domain:
Domain:
Domain:
Range:
Range:
Range:
Explain:
Explain:
Explain:
Evaluate each function for the given value.
41. f(3) if f(x) = 2x + 3
43. h(0.5) if h(x) =
1
x
42. g(- 2) if g(x) = 5x2 + 3x – 2
44. j(2a) if j(x) = 1 – 4x3
45. f(n – 1) if f(x) = 2x2 – x + 9
2
Pre-Calculus Summer Assignment Page 9
51. You can use the table feature of a graphing calculator to find the domain of function. Enter the function into the Y =
list. Draw the sketch. Then observe the y-values in the table. An error indicates that an x-value is excluded from the
domain. Determine the domain of each function.
a. f(x) =
3
b. g(x) =
x 1
3 x
5 x
c. h(x) =
x 2 12
x2 4
Sketch:
Sketch:
Sketch:
Domain:
Domain:
Domain:
52. Education The table shows the number of students who applied and the number of students attending selected
universities.
University
Number Applied
Number Attending
Auburn University
13,264
4,184
University of California-Davis
27,954
4,412
University of Illinois-Urbana-Champaign
21,484
6,366
University of Maryland- College Park
23,117
3,912
State University of New York- Stony Brook
16,849
2,415
The Ohio State University
19,563
5,982
Texas A&M University
17,284
6,949
a. State the relation of the data as a set of ordered pairs. Also state the domain and range of the relation.
b. Determine whether the relation is a function. Explain.
3
Pre-Calculus Summer Assignment Page 23
Read and study the lesson to answer each question.
1. Explain the significance of m and b in y = mx + b.
2. Name the zero of the function whose graph is shown at the right. Explain how you found the zero.
3. Describe the process you would use to graph a line with a y-intercept of 2 and a slope of -4.
4. Compare and contrast the graphs of y = 5x + 8 and y = -5x + 8.
Graph each equation using the x-and y-intercepts.
5.
3x 4 y 2 0
Calculations:
6. x 2y 5
0
Calculations:
4
Pre-Calculus Summer Assignment Page 23
Graph each equation using the y-intercept and the slope.
7. y
x 7
8. y
5
Find the zero of each function. If no zero exists, write none. Then graph the function.
9. f ( x)
1
x 6
2
10. f (x) 19
5
Pre-Calculus Summer Assignment Page 23
11. Archeology Archaeologists use bones and other artifacts found at historical sites to study a culture. One analysis
they perform is to use a function to determine the height of the person from a tibia bone. Typically a man whose tibia is
38.500 centimeters long is 173 centimeters tall. A man with a 44.125-centimeter tibia is 188 centimeters tall.
a. Write two ordered pairs that represent the function.
b. Determine the slope of the line through the two points.
c. Explain the meaning of the slope in this context.
32. Write a linear function that has no zero.
Then write a linear function that has infinitely many zeros.
34. Critical Thinking A line passes through A(3,7) and B(-4,9). Find the value of a if C(a,1) is on the line.
35. Chemistry According to Charles’ Law, the pressure P in pascals of a
fixed volume of a gas is linearly related to the temperature T in degrees
Celsius. In an experiment, it was found that when T = 80, P = 100.
a) What is the slope of the line containing these points?
b) Explain the meaning of the slope in this context.
c) Graph the function.
37. Accounting A business’s capital costs are expenses for things that last more
than one year and lose value or wear out over time. Examples include equipment,
buildings, and patents. The value of these items declines, or depreciates over
time. One way to calculate depreciation is the straight-line method, using the
value and the estimated life of the asset. Suppose v(t) = 10,440 – 290t describes
the value v(t) of a piece of software after t months.
a) Find the zero of the function. What does the zero represent?
b) Find the slope of the function. What does the slope represent?
c) Graph the function.
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Pre-Calculus Summer Assignment Page 29
Read and study the lesson to answer each question.
1. List all the different sets of information that are sufficient to write the equation of a line.
2. Demonstrate two different ways to find the equation of the line with a slope of
1
passing through the point at (3,
4
-4).
3. Explain what 55 and 49 represent in the equation c
service call lasting h hours.
55h
49, which represents the cost c of a plumber’s
4. Write an equation for the line whose graph is shown at the right.
Write an equation in slope-intercept form for each line described.
6. Slope =
1
y-intercept = -10
4,
8. passes through (5, 2) and (7, 9)
7. Slope = 4, passes through (3, 2)
9. Horizontal and passes through (-9, 2)
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Pre-Calculus Summer Assignment Page 29
10. Botany Do you feel like every time you cut the grass it needs to be cut again right away? Be grateful you’re not
cutting the Bermuda grass that grows in Africa and Asia. It can grow at a rate of 5.9 inches a day! Suppose you cut a
Bermuda grass plant to a length of 2 inches.
a. Write an equation that models the length of the plant y after x days.
b. If you didn’t cut it again, how long would the plant be in one week?
c. Can this rate of growth be maintained indefinitely? Explain.
27. Transportation The mileage in miles per gallon (mpg) for city and highway driving
of several recent model-year cars are given.
a.
Find a letter equation that can be used to find a car’s highway mileage based on
its city mileage.
Model
A
B
C
D
E
F
G
H
City
(mpg)
24
20
20
20
23
24
27
22
Highway
(mpg)
32
29
29
28
30
30
37
28
b. Model J’s city mileage is 19 mpg. Use your equation to predict its highway mileage.
c. Highway mileage for model J is 26 mpg. How well did your equation predict the mileage? Explain.
8
Pre-Calculus Summer Assignment Page 35
Using slopes determine whether the graphs of each pair of equations are parallel, coinciding, perpendicular, or none of
these. Show work where necessary.
y
y
5x 5
5x 2
y
x 6
x
y 8
5.
7.
0
y
6.
8.
y
y
6x 2
1
x 8
6
2x 8
4 x 2 y 16
0
9. Write the standard form of the equation of the line that passes through A(5, 9) and is parallel to the graph of
y
5x 9.
10. Write the standard form of the equation of the line that passes through B(-10, -5) and is perpendicular to the graph of
6x 5 y
24.
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Pre-Calculus Summer Assignment Page 35
29. The equation of line m is 8x 14 y
3 0.
a) For what value of k is the graph of kx
b) What is k if the graphs of m and kx
7y 10 0 parallel to line m?
7y 10 0 are perpendicular?
30. Critical Thinking Write equations of two lines that satisfy each description.
a) perpendicular and one is vertical
b) parallel and neither has a y-intercept
31. Geometry An altitude of a triangle is a segment that passes through one vertex and is perpendicular to the opposite
side. Find the standard form of the line containing each altitude of ∆ ABC.
10
Pre-Calculus Summer Assignment Page 41
Read and study the lesson to answer each question.
2. Describe the different methods for finding a best-fit line for a set of data.
3. Write about a set of real-world data that you think would show a negative correlation.
5. Education Do you share a computer at school? The table shows the average number of students per computer in
public schools in the United States.
Students per Computer
Academic 1988- 1989- 1990- 1991- 1992- 1993- 1994- 1995- 1996- 1997- 1998- 1999- 2000- 2001Year
1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Average
25
22
20
18
16
14
10.5
10
7.8
6.1
5.7
5.4
5
4.9
Complete parts a through d for each set of data given in exercise 5.
a) Graph the data on a scatter plot.
b) Use two ordered pairs to write the
equation of a best-fit line.
c) Use a graphing calculator to find an
equation of the regression line for the data.
What is the correlation of the coefficient?
d) If the equation of the regression line
shows a moderate or strong relationship, predict
the missing value. Explain whether you think
the prediction is reliable.
11
?
1
Pre-Calculus Summer Assignment Page 41 & 54
13. Critical Thinking This table shows the median salaries of American men and women for several years. According to
the data, will the women’s median salary ever be equal to the men’s? If so, predict the year. Explain.
Men’s
20,293
20,469
20,455
21,102
21,720
22,562
Year
1990
1991
1992
1993
1994
1995
Women’s
10,070
10,476
10,714
11,046
11,466
12,130
Year
1996
1997
1998
1999
2000
2001
Men’s
23,834
25,212
26,492
27,275
28,343
29,101
Women’s
12,815
13,703
14,430
15,311
16,063
16,614
Page 54
Read and study the lesson to answer each question.
1. Write the inequality whose graph is shown.
Graph each inequality.
4. x
y
4
5. 3x
y
6
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Pre-Calculus Summer Assignment Page 54
6. 7
x
y
9
7. y
x
3
8. Business Nancy Stone has a small company and has negotiated a special rate
for rental cars when she and other employees take business trips. The maximum
charge is $45.00 per day plus $0.40 per mile. Discounts apply when renting for
longer periods of time or during off-peak seasons.
a) Write a linear inequality that models the total cost of the daily rental
c(m) as a function of the total miles driven, m.
b) Graph the inequality.
c) Name three combinations of miles and total cost that satisfy the
inequality.
13
Chapter 2 Pre-Calculus Summer Assignment Page 70
Read and study the lesson before answering each question.
1. Write a system of equations in which it is easier to use the substitution method to solve the system rather than the
elimination method. Explain your choice.
State whether each system is consistent and independent, consistent and dependent, or inconsistent.Show all work
needed.
11.
13.
x 3y
x 2y
18
7
35x 40 y
7x
12.
y
2y
0.5 x
x 4
55
8 y 11
Solve each system of equations by graphing.
14.
x 5
4x 5y
20
17.
x 3y
2x 6 y
0
5
14
Pre-Calculus Summer Assignment Page 70
Solve each system of equations algebraically.
21.
5x
y 16
2x 3y
3
x
5
28.
1
x
5
1
y
6
5
y
6
26.
3
x
2x
2y 8
y
7
1
11
32. Sports Spartan Stadium at San Jose State University in California has a seating capacity of about 30,000. A
newspaper article states that the Spartans get four times as many tickets as the visiting team. Suppose S represents the
number of tickets for the Spartans and V represents the number of tickets for the visiting team’s fans.
a) Which system could be used by a newspaper reader to determine how many tickets each team gets?
A)
4 S 4V
S 4V
30,000
B)
S 4V 0
S V 30,000
C)
S V 30,000
V 4S 0
c) Solve the system to find out how many tickets each team gets.
15
Pre-Calculus Summer Assignment Page 70
33. Geometry Two triangles have the same perimeter of 20 units. One triangle is an isosceles triangle. The other
triangle has a side 6 units long and its other two sides are the same lengths as the base and leg of the isosceles triangle.
a) What are the dimensions of each triangle?
b) What type of triangle is the second triangle?
16
Lesson 10.3 Read and Review
GOAL: Use special product patterns for the product of a sum and a difference and for the square of a binomial and sue special
products in real-life models
VOCABULARY
Some pairs of binomials have special product patterns as follows.
EXAMPLE 1 Using the Sum and Difference Pattern
Sum & Difference Pattern
Use the sum and difference pattern to find the product
(a
b)(a
b)
a2
(4 y 3)(4 y 3).
b2
Square of a Binomial Pattern
( a b) 2
a2
2ab b 2
( a b) 2
a2
2ab b 2
SOLUTION
For this trinomial, b = 6 and c = 8. You need to find two numbers whose sum is 6 and whose product is 8.
(a
b)(a
(4 y
a2
b)
3)(4 y
Write pattern.
(4 y ) 2
3)
16 y 2
b2
32
Apply pattern.
9
Simplify.
Exercises for Example 1
Use the sum and difference pattern to find the product.
1. ( x 5)( x 5)
EXAMPLE 2 Squaring a Binomial
(3x 2)(3x 2)
2.
3.
( x 2 y)( x 2 y)
Use the square of a binomial pattern to find the product.
a.
(2 x
3) 2
b. ( 4 x
1) 2
SOLUTION
a.
(a
b) 2
( 2 x 3)
a2
2
2ab b 2
(2 x )
2
Write pattern.
2
2( 2 x )(3) 3
4 x 2 12x 9
b.
(a
b) 2
( 4 x 1) 2
16x 2
a2
8x 1
Simplify.
2ab b 2
(4 x ) 2
Apply pattern.
2( 4 x )(1) 12
Write pattern.
Apply pattern.
Simplify.
17
Practice: Complete all circled exercises on white lined paper.
Write the product of the sum and difference.
1.
( x 5)( x 5)
3. (3x
5)(3x 5)
5. (5n
5)(5n 5)
2. (t
4)(t 4)
4. (7 x
6)(7 x 6)
14. (x
9) 2
Write the square of the binomial as a trinomial.
13. (x
16. ( 2m
19. (b
3) 2
3) 2
2 2
)
3
17. (7 y
3) 2
20. (m
1 2
)
2
Find the product.
25. (c
5)(c 5)
31. (3 y
2 x)2
1
3
28. ( x
1
6)( x 6)
3
18
Lesson 10.5 Read and Review
GOAL: Factor a quadratic expression of the form
x2
bx c and solve quadratic equations by factoring
EXAMPLE 1 Factoring when b and c are Positive
VOCABULARY
To factor a quadratic expression means to
write it as the product of two linear
expressions. To factor x 2 bx c
you need to find numbers p and q such
that
p q b and pq
Factor
x2
6 x 8.
SOLUTION
For this trinomial, b = 6 and c = 8. You need to find two numbers whose sum is 6
and whose product is 8.
x2
c.
6x 8
(x
p )( x
q) Find p and q when p + q = 6 and pq = 8
( x 4)( x 2)
p = 4 and q = 2
Exercises for Example 1
Factor the trinomial.
x2
2.
5x 6
2.
x2
6x 5
3.
x2
3x 2
EXAMPLE 2 Factoring when b is Negative and c is Positive
Factor
x2
5x 4.
SOLUTION Because b is negative and c is positive, both p and q must be negative numbers. Find two numbers whose sum is
-5 and whose product is -4.
x2
5x
4
(x
p )( x
q)
( x 4)( x 1)
Find p and q when p + q= -5 and pq = 4
p = - 4 and q = --1
Exercises for Example 2
Factor the trinomial.
4. x 2
3x 2
5. x 2
7 x 12
6. x 2
5x 6
19
EXAMPLE 3 Factoring when b and c are Negative
Factor
x2
3x 10.
SOLUTION For this trinomial, b = -3 and c = - 10. Because c is negative, you know that p and q cannot both have negative values.
x2
3x 10
(x
p )( x
q)
Find p and q when p + q = -3 and pq = -10.
( x 2)( x 5)
p = 2 and q = -5
Exercises for Example 3
Factor the trinomial.
7.
x2
x 2
8.
x2
4 x 12
EXAMPLE 4 Solving a Quadratic Equation
SOLUTION:
Exercises for Example 4
Solve the equation.
10.
11.
x 2 8x 15 0
x 2 8x 12 0
9.
Solve
x2
x
x
x
x
6
4 x 12.
Write in standard form.
0 or (x 2)
0
Factor the left side. Because c is negative, p
and q cannot both have negative values; p =
6 and q = -2
Use zero-product property.
Set first factor equal to 0.
Solve for x.
Set second factor equal to 0.
Solve for x.
The solutions are -6 and 2.
0
6
2
2
2x 8
Write the equation.
x 2 4 x 12
x 2 4 x 12 0
( x 6)( x 2) 0
(x 6)
x2
0
12.
x2
3x 4 0
20
Practice:
Match the trinomial with a correct factorization.
1.
2.
3.
4.
5.
6.
x 2 5x 6
x 2 5x 6
x2 x 6
x2 x 6
x 2 4x 4
x 2 6x 9
A. ( x
B. ( x
C. ( x
2)( x
3)( x
3)( x
D. ( x 3)( x
E. ( x 3)( x
F. ( x 3)( x
2)
2)
3)
2)
2)
2)
Factor the trinomial.
x 6
8. x 2
10. x 2
5x 4
11. x 2
x 42
13. x 2
16x 64
14. x 2
13x 36
7.
x2
8x 15
Solve the equation by factoring.
16.
x2
3x 40 0
17.
x 2 16x 63 0
18.
x 2 11x 28 0
21
Lesson 10.7 Read and Review
GOAL: Use special product patterns to factor quadratic polynomials and solve quadratic equations by factoring
VOCABULARY
Factoring Special Products
Difference of Two Squares Pattern
Example
a2
9x2
b2
(a
b)(a
b)
16
(3x
4)(3x
4)
Perfect Square Trinomial Pattern
Example
a2
2ab b 2
(a
b) 2
x2
18x 16
(x
4) 2
a2
2ab b 2
( a b) 2
x2
12 x
(x
6) 2
36
EXAMPLE 1 Factoring the Difference of Two Squares
a.
n2
25
b. 4 x
2
y2
SOLUTION
a.
n2
25 n 2
52
Write as
4x2
y2
(2 x ) 2
(2 x
y2
y)(2 x
b2 .
Factor using difference of two squares pattern.
(n 5)(n 5)
b.
a2
Write as
y)
a2
b2 .
Factor using difference of two squares pattern.
Exercises for Example 1
Factor the expression.
1.
16 9 y 2
2. 4q
2
49
3.
36 25x 2
22
EXAMPLE 2 Factoring Perfect Square Trinomials
x2
a.
6x 9
b. 9 y
2
12 y
4
SOLUTION
x2
a.
6x
x2
9
2( x )(3) 32
3) 2
(x
9y2
b.
12 y
4
Write as
(3 y ) 2
(3 y
a2
2ab b2 .
Factor using perfect square trinomials.
2(3 y )(2) 2 2
Write as
2) 2
a2
2ab b2 .
Factor using perfect square trinomial pattern.
Exercises for Example 2
Factor the expression.
4.
x 2 18x 81
5.
4n 2
2.
121 x 2
4.
9x 2
4y2
6.
49 x 2
1 2
y
9
16.
20n 25
6. 16 y
2
8y 1
Practice: Factor the expression.
x2
1.
3.
9 4x2
5. 25x
7.
9
2
1 2
x
4
49
1
4
x 2 14x 49
23
17. 4 x
19.
2
4 xy
y2
25 10x x 2
21. 9 x
2
12 xy
4y2
18.
9 x 2 12x 4
20.
9 12x 4 x 2
22.
x2
3
9
x
2
16
Use factoring to solve the equation. Use a graphing calculator to check your solution if you wish.
31.
18x 2 50 0
32.
5x 2
40x 80 0
24