Semester 2 Final Review
Algebra
7-1 Find an exponential function in y = abx form that satisfies each of the following sets of conditions.
a. Has a y-intercept of (0, 2) and a multiplier of 0.8.
b. Passes through the points (0, 3.5) and (2, 31.5)
7-2 Sam wants to create an arithmetic sequence and a geometric sequence, both of which have t(1) = 8
and t(7) = 512. Is this possible? If it is, help sam create his sequences. If not, justify why not.
7-3 Write an equation or system of equations to solve this problem.
An adult ticket to the amusement park costs $24.95 and a child’s ticket costs $15.95. A group of 10 people
paid $186.50 to enter the park. How many were adults?
7-4 Write each expression below in radical form and compute the value without using a calculator.
a. 81/3
b, 163/4
c. 1254/3
7-5 A share of ABC stock was worth $60 in 2005 and only worth $45 in 2010.
a. Find the multiplier and the percent decrease.
b. Write an exponential function that models the value of the stock starting from 2005.
c. Assuming that the decline in value continues at the same rate, use your answer to (b) to predict the value in
2020.
7-6 Solve each system of equations.
a.
b.
7-7 Below are several situations that can be described using exponential functions. They represent a small
sampling of the situations where quantities grow or decay by a constant percentage over equal periods of
time. For each situation (a) through (d):

Find an appropriate unit of time (such as days, weeks, years).

Find the multiplier that should be used.

Identify the initial value.

Write an exponential equation in the form y = abx that represents the growth or decay.
a. A house purchased for $120,000 has an annual appreciation of 6%.
b. The number of bacteria present in a colony is 180 at noon, and it increases at a rate of 22% per hour.
c. The value of a car with an initial purchase price of $12,250 depreciates by 11% per year.
d. An investment of $1000 earns 6% annual interest, compounded monthly.
7-8 Write an equation for the line that passes through the points (–5, 4) and (3, –2).
7-9 Marissa looked at a different effect of the weather. She studied a linear association between the
attendance at the amusement park and the temperature. Marissa made
the residual plot at right.
a. Was a linear model appropriate? Why or why not?
7-10 Mary helps prepare food in the Tiger Café. Mary notices that sales of fresh fruit cups seem to vary widely
from day to day. This is a problem because preparing too many cups results in wasted fruit and making too
few results in lost sales. She decides that the daily weather may have a strong association with demand. Mary
chooses 12 days at random from last semester and pairs each day’s high temperature with the number of
fresh fruit cups sold each day.
Create a linear model for this data by finding the LSRL. Sketch the graph and the LSRL.
a. Describe the association. Make sure you describe the form and provide evidence for the form. Provide
numerical values for direction and strength and interpret them in context. Describe anyoutliers.
b. If Mary wanted to be reasonably certain of not running out of fresh fruit cups on a day forecasted to be 90
degrees, how many should she prepare? (Hint:Consider the upper and lower bounds of sales.)
8-1 Factor and use the Zero Product Property to find the roots of the following quadratic equations.
a. 0 = x2 − 7x + 12
b. 0 = 6x2 − 23x + 20
c. 0 = x2 − 9
d. 0 = x2 + 12x + 36
8-2 The price of milk has been steadily increasing 5% per year. If the cost of a gallon is now $3.89:
a. What will it cost in 10 years?
b. What did it cost 5 years ago?
8-3 Without using a calculator, simplify using only positive exponents.
a.
b.
c.
8-4 Use the graph at left to answer the questions below.
e. a. One of these lines represents Feng, and one represents Wai. Write an
equation for each girl’s line.
b. The two girls are riding bikes. How fast does each girl ride?
c. When do Feng and Wai meet? At that point, how far are they from
school?
8-5 Graph y = x2 − 2x. Identify the roots, y-intercept, x-intercepts, and the vertex.
8-6 Find the coordinates of the y-intercept and x-intercepts of y = x2 − 2x − 15.. Show all of the work that you
used to find these points.
8-7 Given the two points (−24, 7) and (30, 25),
a. What is an equation of the line passing through the points?
b. Is (51, 33) also on the same line? Explain your reasoning?
8-8 Write the equation of the following two sequences in “first term” form.
a. 100, 10, 1, 0.1, …
b. 0, –50, –100, …
8-9 Write the first four terms of the following sequences.
a. an = 3 · 5n−1,...
b. a1 = 10, an + 1 = − 5an
9-1 Write an inequality that represents the graph at right.
9-2 Is the point (0, 4) a solution to the system of inequalities below? Justify your answer.
9-3 Brian was holding a ballroom dance. He wanted to make sure girls would come, so he charged boys $5 to
get in but girls only $3. The 45 people who came paid a total of $175. How many girls came to the dance?
9-4 Factor these quadratic expressions completely, if possible.
a. x2 + x – 30
b. −3x2 + 23x2 − 14x
c. 2x2 − 5x + 4
d. 6x3 + 10x2 − 24x
9-5 Solve each inequality below for the given variable.
a. 4x − 3 ≥ 9
b. 3(t + 4)< 5
c.
9-6 Solve each quadratic equation using the specified method.
b. The Quadratic Formula
0 = 3x2 + 4x − 7
c. Factoring
x2 − 3x − 18 = 0
d. Completing the square
x2 + 4x + 1 = 0
e. Using a graph
2x2 + 5x − 12 = 0
9-7 Given the quadratic function f(x) = (x − 1)2 − 4:
a. State the location of the vertex.
b. Determine the x-intercepts.
d. 5x + 4 > − 3(x − 8)
c. Sketch a graph of the function.
9-8 Graph the system of inequalities below on graph paper.
9-9 Lew says to his granddaughter Audrey, “Even if you tripled your age and added 9, you still wouldn’t be as
old as I am.” Lew is 60 years old. Write and solve an inequality to determine the possible ages Audrey could
be.
9-10 The cost to rent a DVD has decreased 10% per year over the past several years.
a. If the current cost is $5, write an exponential equation describing this situation.
b. According to the equation, what did it cost to rent a DVD 5 years ago?
9-11 Find the equation of an exponential function of the form y = abx that passes through the points:
a. (3, 13.5) and (5, 30.375)
b. (5, 24.41) and (12, 5.58)
10-1 Solve the equations below using any method. How many solutions does each problem have?
a.
b.
c.
d.
10-2 Lately there have been a number of times when the sound quality of the news interviews on the school’s
video station has been unfit to broadcast. One source of the sound problem might be that one or two of the
microphones is not working well. Brendan collected the following data from the last broadcast season:
a. What is the probability that a randomly selected show will have an unacceptable sound quality?
a. Is there an association between the sound quality and the microphone used? Should Brendan keep
searching for the source of the sound problem or has he found it?
10-3 For the exponential function y = 20(1.06)x:
a. What are the starting point (y-intercept) and multiplier? Sketch a graph.
b. This function describes the yearly growth of a bank account. What percent interest does the account
earn per year?
10-4 Factor each polynomial.
a. x2 − x – 56
b. 3x2 − 4x + 1
c. 2x3 + x2 + x
d. 2x2 − 50
10-5 Solve each quadratic equation using any method. Describe the answers as rational or irrational
numbers.
a. 2x2 − x − 5 = 0
b. 4x2 = 4x – 1
10-6 For an exponential function of the form f(x) = abx, f(3) = 24 and f(4) = 48. Find an equation that matches
this data.
10-7 Use a graph to estimate the solutions to 2x = x + 3.
10-8 For the quadratic function f(x) = (x − 3)2 + 4:
a. Identify the vertex and tell if it the maximum or minimum point of the function.
b. Why does (x − 3)2 + 4 = 0 have no real solutions?
11-1 Consider the quadratic inequality
.
a. Solve for the boundary point(s). How many boundary points are there?
b. Place the boundary point(s) on a number line. How many regions do you need to test?
c. Test each region and determine which one(s) make the inequality true. Identify the solution algebraically
and on the number line.
[ a: x = 5, –10, two of them; b: Three regions; c: –10 < x < 5 ]
11-2 Consider the quadratic inequality
.
a.
Solve for the boundary point(s). How many boundary points are there?
b.
Place the boundary point(s) on a number line. How many regions do you need to test?
c.
Test each region and determine which one(s) make the inequality true. Identify the solution
algebraically and on the number line.
[ a: x = 9, –6, two of them; b: Three regions; c: –6 < x < 9 ]
11-3 Consider the quadratic inequality x2 + 2x – 3  12. Find the boundary points and show all of the
solutions on a number line. [ x  –5 or x  3 ]
11-4 Solve the inequalities below and represent your solutions on a number line.
a.
x6 7
b.
2x  5  0
[ a: 13  x  1 , b: all real numbers ]
11-5 In a survey of 200 pet owners, 76 claimed to not own a cat and 63 indicated they did not own a dog.
Eighty–three responded that they own both a dog and a cat. Make a two-way table displaying the counts in
each category. Is there an association between cat and dog ownership? Be sure to explain your reasoning by
showing the probability of each.
11-6 Is there an association between Parker’s pitching style and Brandon’s chance of getting to base when
batting in softball games? Brandon batted 32 times when Parker was pitching. Brandon got to base 8 times.
Parker used his fastball 12 times total, but Brandon got to base only three times when Parker used his fastball.
Make a two-way table and decide whether there is an association between pitching style and getting to base.
11-7 Solve the following inequalities algebraically. Give your answer as an inequality..
a. y  x 2  3x  1
b. y  x  2
c. y  x  1
11-8 Draw a graph of the solution region for the system of inequalities below.
yx4
y
2
y   54 x  6
2
–2
[ Solution graph is at right. ]
11-9 Determine the number of solutions for each quadratic equation below by first completing the square.
Explain how you can determine the number of solutions once the equation is written in perfect square form.
a.
x2 + 6x + 8 = 0
b.
x2 – 6x + 9 = 0
[ a: (x + 3)2 = 1, two solutions; b: (x – 3)2 = 0, one solution ]
More Review
1. Solve by using any method.
a.
t 2  8t  10  0
b.
c.
x2 + 6x + 8 = 0
d.
[ a: 4  6 , b: 3  8 c: x = –4 and –2, d: x = 0 and –2 ]
p2  6 p  1  0
3x2 + 6x = 0
x
2. Decide whether each of the following points is on the graph of y = –2x2 + 5x – 1. Explain how you know.
a.
(–2, –19)
b.
(2, 19)
[ a: yes, b: no ]
3. Solve 100  x 2  2  x .
[x=6]
4. Aaron just inherited $8000 from his grandmother. He plans to invest the money, not touching it for twenty
years. He can invest in treasury bills at 6.67% interest, compounded annually, or in a money market account
earning 6.5% annual interest, compounded weekly. Which should he do? Justify your answer.
[ The money market account will yield a little more. ]
5. Factor each expression below.
a.
9x2 + 24xy + 16y2
b.
x3 + 4x2 + 4x
[ a: (3x + 4y)(3x + 4y), b: (x)(x + 2)(x + 2)
6. This parabola shows an equation which could be considered a “special quadratic”. What equation is it and
why is it considered a special quadratic?
[ y=(x–2)(x–2) perfect square trinomial; there is only one root ]
7. This parabola shows an equation which could be considered a “special quadratic”. What equation is it and
why is it considered a special quadratic?
[ y=(x–3)(x+3) or y= x2-9 : difference of squares; there is no middle term ]
8. One type of “special quadratic” is called a “perfect square trinomial”. Why is this considered a “special
quadratic”? Refer to both the equation itself and the graph of the parabola in your answer.
[ A perfect square trinomial is the square of a binomial and it only has one root or x-intercept ]
9. One type of “special quadratic” is called a “difference of squares”. Why is this considered a “special
quadratic”? Refer to both the equation itself and the graph of the parabola in your answer.
[ A difference of squares quadratic is always in the form of y=(a+b)(a–b) and in sum form y= a2–b2. The
graph of this type of parabola will have roots in which the absolute values of the roots are equal. (5 and –5,
for example) ]
10. Antonio is president of Maths-R-Us and has asked his financial advisor for a report about the company’s
expected profits. The advisor told him that the profits were expected to follow the equation y =
150210(0.82)x over the next 12 months. Tell Antonio everything you can tell about the company from the
equation.
[ Sample statements: They begin with $150,210.00 expected profit, which is decreasing at a rate of 18% per
month. At the end of the year, they will have $13,882.42 expected profit. ]
11. Write an equation of an exponential function that passes through the points (0, 2) and (2, 18). Using a
table and graph may be useful.
[ y  2  3x ]
12. Each table below represents an exponential function of the form y  ab x . Copy and complete each table
on your paper and find the corresponding rule.
a.
b.
x
y
x y
0 3.1
0 2
1 4.34
1 10
2 6.076
2
3
13. Build a rectangle for each of the following expressions using Algebra Tiles. Then sketch the rectangle on
your paper. Find its dimensions and write its area as a product and a sum.
a.
3x2 + 3x
b.
x2 + xy + y + 3x + 2
c.
2x + 2 + 2xy + 2y + 2x + 2
d.
x2 + 4x + 5
[ a: 3x2 + 3x = 3x(x + 1), b: x2 + xy + y + 3x + 2 = (x + 1)(y + x + 2), c: 2xy + 2y + 4x + 4 = (2x + 2)(y + 2), d: x2 +
4x + 5 cannot be made into a rectangle because there is one too many units (needs 4 instead of 5) ]
14. Use the table below to write a quadratic rule. Explain how you created your equation.
x
y
–7
0
–6
–5
–5
–8
–4
–9
–3
–8
–2
–5
–1
0
0
7
1
16
2
27
[ Answers will vary. Students should explain and show steps that reverse the Zero Product Property to
come up with the rule y  (x  7)(x  1) or y  x 2  8x  7 . ]
15. Jamie threw a softball that traveled along a path described by the parabola y = -x2 + 10x, where y = the
height of the softball in feet and x = the horizontal distance in feet that the ball has traveled from Jamie.
On separate graph paper, graph the path of the softball.
a.
Where does the softball hit the ground? How can you tell?
b.
Find the vertex. What information does the vertex tell you?
c.
What is the ball’s horizontal distance from Jamie when it is 24 feet up in the air? Does this
make sense? Explain.
d.
Does any of your data not make sense? Explain.
[ a: 10 feet away from her, b: (5, 25); the ball is at its highest point of 25 feet when it is a horizontal distance
of 5 feet from Jamie; c: at 4 feet and at 6 feet [the points (4, 24) and (6, 24)], yes, it makes sense, because
the ball is 24 feet off the ground two times, on the way up and on the way down; d: the points beyond x =
10 don’t make sense because the represent negative (below ground) height ]