Math 107 Study Guide for Chapters 3 and 4
PRACTICE EXERCISES
1. Find the function whose graph is a parabola with vertex (−3,1) and that passes through the point
(5, −15) in vertex form: 𝑓 (𝑥) = 𝑎(𝑥 − ℎ )2 + 𝑘. Sketch the graph on the axes below and label the
vertex and any intercepts on the graph.
Vertex form: _________________________
2. Find the equation of the quadratic function whose graph is shown.
y
Vertex form: ___________________________
x
3. A 5.5 feet tall woman is shooting a free throw. The path of the basketball is parabolic in shape and
the ball reaches its maximum height of 11.5 feet when the ball is 10 feet from the player.
(a) Find the equation for the path of the ball. Let 𝑥 be the horizontal distance from the shooter and
𝑦 be the height of the ball. (Assume that the ball is 5.5 ft. high when it is released.)
(b) The ball hits the front of the rim, which is 10 feet high. How far is the shooter from the rim?
4. After experimentation, two college students find that when a bottle of Washington Champagne is
shaken several times, held upright and uncorked, its cork travels according to the function defined
by 𝑠(𝑡) = −16𝑡 2 + 64𝑡 + 3 where 𝑠 is the cork’s height in feet above the ground and 𝑡 is the time
in seconds after the cork has been released.
(a) After how many seconds will the cork reach its maximum height?
(b) What is the maximum height of the cork?
1
8
5. The path of a paper airplane is y = − x +
2
1
x + 4 , where 𝑦 is the height in feet from the ground and
2
𝑦 is the horizontal distance in feet from the person throwing the paper airplane.
(a) How high off of the ground is the paper airplane when it is released from the thrower’s hand?
(b) What is the horizontal distance from the thrower that the airplane will be when it reaches its
maximum height?
(c) What is the maximum height that the paper airplane will reach?
(d) How far from the thrower will the airplane be when it hits the ground?
6. A gardener has 80 feet of fencing to fence in a rectangular garden. He wants to put the fence all the
way around the garden, and he wants to section it into 4 areas with fencing parallel to one side of the
rectangle, as shown in the picture below.
(a) Find a function that models the total area of the garden that he can fence in terms of its length 𝑥.
x

y
(b) Find the dimensions of the largest possible total area that he can fence.
7. Consider the polynomial function 𝑃(𝑥) = −3(𝑥 + 2)2 (𝑥 − 4)(𝑥 + 5)3 .
(a) What is the degree of the polynomial 𝑃(𝑥)?
(b) Determine the end behavior of P ( x) .
(c) Find the zeros of 𝑃(𝑥) and determine the multiplicity of each.
(d) Sketch a graph of 𝑃(𝑥). Label any intercepts on the graph.
8. Consider the graph of the polynomial function given below. Which of the following statement(s) is
(are) true? Circle all possible true statements. Explain your reasoning.
a. The polynomial could have degree 4.
30
20
b. The polynomial could have degree 5.
10
c. The polynomial could have degree 6.
-2
2
-10
d. The polynomial could have degree 7.
-20
e. The polynomial could have degree 8.
-30
Reasoning:
_____________________________________________________________________________
_____________________________________________________________________________
9. Determine a polynomial 𝑃 with real coefficients which satisfies the given conditions. If no such
polynomial exists, explain why. You do not need to multiply out the polynomial.
(a) 𝑃 has degree 4, leading coefficient 2, and zeros 3 (multiplicity 3) and 1 + 𝑖 (multiplicity 1).
(b) 𝑃 has degree 5, leading coefficient -2, and zeros 1 (multiplicity 1) and −5 − 𝑖 (multiplicity 2).
(c) 𝑃 has degree 3, zeros 2 (multiplicity 1) and −5 − 𝑖 (multiplicity 1), and has constant term of 26.
10. Suppose the polynomial 𝑝(𝑥) = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥 2 + ⋯ + 𝑎9 𝑥 9 has real coefficients with 𝑎9 ≠ 0.
Suppose also that it is known that 2, 8, and 1 − 3𝑖 are zeros of the polynomial 𝑝. Using this
information, answer the following questions. Justify your answer.
(a) What is another zero of 𝑝?
(b) At most how many real zeros can 𝑝 have?
(c) At most how many complex (non-real) zeros can 𝑝 have?
11. (Text p. 268 #23-#28) For each
polynomial graph
(a) state wether the degree of the
function is even or odd;
(b) use the graph to name the
zeros of 𝑓, then state whether
their multiplicity is even or odd;
(c) state the minimum possible
degree of 𝑓 and write it in
factored form; and
(d) estimate the domain and range.
Assume all zeros are real
12. (Text p. 268 #37-#42) Every function
below has zeros 𝑥 = −1, 𝑥 = −3, and
𝑥 = 2. Match each to its corresponding
graph using degree, end behavior, and
multiplicity of each zero.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
𝑓1 (𝑥) = (𝑥 + 1)2 (𝑥 + 3)(𝑥 − 2)
𝑓2 (𝑥) = (𝑥 + 1)(𝑥 + 3)2 (𝑥 − 2)
𝑓3 (𝑥) = (𝑥 + 1)(𝑥 + 3)(𝑥 − 2)3
𝑓4 (𝑥) = (𝑥 + 1)3 (𝑥 + 3)(𝑥 − 2)
𝑓5 (𝑥) = (𝑥 + 1)2 (𝑥 + 3)(𝑥 − 2)2
𝑓6 (𝑥) = (𝑥 + 1)3 (𝑥 + 3)(𝑥 − 2)2
13. Given the polynomial function 𝑔(𝑥) = 3𝑥 7 + 4𝑥 6 − 21𝑥 5 − 10𝑥 4 + 24𝑥 3 , find the following.
(a) Right-End Behavior: As x → ∞ ,
Circle one: 𝑔(𝑥) → ∞ (the function points up) or 𝑔(𝑥) → ∞ (the function points down). Why?
(b) Left-End Behavior: As x → −∞ ,
Circle one: 𝑔(𝑥) → ∞ (the function points up) or 𝑔(𝑥) → ∞ (the function points down). Why?
(c) List all possible zeros of 𝒈(𝒙).
(d) Find all zeros (real and complex) of 𝒈(𝒙) by factoring. List all zeros and identify any multiplicity.
(e) Sketch the graph of 𝒈(𝒙). Label any intercepts on the graph.
14. Answer the following for g ( x) = x 3 − 5 x 2 + 2 x + 12 .
(a) List all possible rational zeros of g ( x) .
(b) Find all the zeros of g ( x) .
(c) Write g ( x) as a product of linear factors.
15. Perform the division: ( x 3 − 4 x + 5) ÷ ( x − 2) .
Write your answer in the form
x3 − 4 x + 5
R( x)
, where Q ( x) is the quotient and R ( x)
= Q( x) +
x−2
x−2
is the remainder of the division.
16. Dividing the polynomial 𝑃(𝑥) by 𝑥 + 3 yields the quotient 𝑄(𝑥) and a remainder of 5. If 𝑄 (2) = 2,
determine 𝑃(−3) and 𝑃 (2).
17. Determine all the asymptotes, intercepts and domain for each of the following functions:
(a) f ( x) =
x+2
5 x 2 − 10
(b) g ( x) =
5 − 4x2
7 x 2 + 15 x + 2
(c) h( x) =
2 x2 − x + 6
x −1
18. Which, if any, of the following rational functions have the same asymptotes and intercepts as the
function shown? Explain your reasoning.
19. For each part, determine a rational function satisfying the given criteria. Show work and explain why
you choose each part.
(a) 𝑈(𝑥) has zeros at 𝑥 = 0 and 𝑥 = −3, vertical asymptotes 𝑥 = 5 and 𝑥 = −2, and horizontal
asymptote 𝑦 = 4.
(b) 𝑉(𝑥) has zeros at 𝑥 = 0 and 𝑥 = −3, vertical asymptotes 𝑥 = 5 and 𝑥 = −2, and horizontal
asymptote 𝑦 = 0.
(c) 𝑊(𝑥), different from your answer to (b), satisfying the same criteria as 𝑉(𝑥).
4𝑥 2−36
20. Let 𝐹 (𝑥) = 2𝑥 2
. Find the following, showing work as to how you found each (even if you can
+7𝑥−9
do it in your head). If necessary, explain how you found an answer. Sketch a graph of 𝐹(𝑥) and
label the asymptotes and intercepts on the graph.
10
(a) Domain in interval
notation
y
8
6
(b) 𝑦-intercept
4
2
(c) 𝑥-intercept(s)
x
-10
-5
5
10
-2
(d) All asymptotes
-4
-6
-8
-10
Now, amend your graph above, so that you are graphing 𝑯(𝒙) =
on the graph specifically.
�𝟒𝒙𝟐 −𝟑𝟔�(𝒙+𝟓)
�𝟐𝒙𝟐 +𝟕𝒙−𝟗�(𝒙+𝟓)
Label the difference
21. Let
Find the following, showing work as to how you found each (even if you can do it
in your head). If necessary, explain how you found an answer. Sketch a graph of
asymptotes and intercepts on the graph.
10
(a) Domain in interval notation
and label the
y
8
6
(b) -intercept
4
(c) -intercept(s)
2
(d) All asymptotes
x
-10
-5
5
10
-2
-4
-6
-8
-10
22. The figure below shows the graph of a rational function with vertical asymptotes
,
,
and horizontal asymptote
. The graph has -intercepts at
and
, and it passes
through the point
. The equation for
has one of the five form shown below. Choose the
appropriate form for
, and then write the equation. Assume that
is in simplest form.
23. Let f and g be quadratic polynomial functions, of the form y = ax 2 + bx + c , where a = 1 , whose
graphs are given below.
y
y
6
4
6
f(x)
4
2
g(x)
2
x
-5
5
Set V ( x ) =
x
-5
5
-2
-2
-4
-4
-6
-6
f (x )
.
g (x )
(a) What is (are) the vertical asymptote(s) of V?
(b) What is (are) the zero(s) of V?
(c) What is the y-intercept of V?
(d) Which of the following is an asymptote of V? (Circle one.)
I. 𝑦 = 0
II. 𝑦 = 1
III. 𝑦 = −1
IV. 𝑦 = 𝑥 + 1
24. A landscaper’s design calls for a dwarf evergreen trees that typically grows rapidly at first, and then
more slowly, to a maximum height of 6 feet. A graph of the height ℎ (in feet) 𝑡 years after sprouting
is shown in the figure below.
(a) Assume that the height approaches 6 feet. If the height of a 2 year old tree is 4 feet, determine
a model of the form h =
at
, 𝑡 > 0 where 𝑎 and 𝑏 are constants.
t +b
(b) Use the model to determine the age of a tree that is 5 feet tall.
(c) Give a practical interpretation of the horizontal asymptote.
25. Match the correct solution to the inequality, 𝑓(𝑥) ≤ 0 where the graph of 𝑓(𝑥) which is given
below.
(a) 𝑥 ∈ (−∞, −1) ∪ (0, 2)
(b) 𝑥 ∈ [0,1] ∪ (2, ∞)
(c) 𝑥 ∈ [−5, −1] ∪ [2,5]
(d) 𝑥 ∈ (−∞, −1) ∪ [0,2)
(e) none of these
26. Solve the inequality inequalities. Express the answer in interval notation.
(a) (𝑥 + 2)3 (𝑥 − 2)2 (𝑥 − 4) ≤ 0
(b)
(c)
2𝑥−𝑥 2
𝑥 2+4𝑥−5
𝑥+1
𝑥−2
<0
𝑥+2
≥ 𝑥+3
27. An airplane manufacturer can produce up to 15 planes per month. The profit made from the sale of
these airplanes is closely modeled by 𝑃 (𝑥) = −0.35𝑥 2 + 4𝑥 − 5, where 𝑃(𝑥) is the profit in
hundred-thousands of dollars per month, and 𝑥 is the number of planes sold.
(a) Find the 𝑦-intercept and explain what it means in this context.
(b) How many airplanes must be sold in order to earn a profit (𝑃 > 0)?
28. A rectangular pen is to be constructed that will enclose 481 square feet.
(a) Determine the length 𝐿(𝑥) of the fence needed as a function of the width 𝑥 of the pen.
(b) Determine the dimensions of the pen if 100 feet of fencing are to be used.
(c) Determine an interval for the possible width of the pen if less than 100 feet of fencing is to be
used.
29. Is the function 𝑓 (𝑥) = 2𝑥 3 + 4 one-to-one? If so, find 𝑓 −1 (𝑥). Sketch the graph of 𝑓 −1 (𝑥).
30. Suppose 𝑓(𝑥) is a one-to-one function. Given that 𝑓(2) = 7 which of the following CANNOT be
true?
(b) 𝑓 −1 (7) = 2
(a) 𝑓 (7) = 2
(c) 𝑓 −1 (5) = 3
(d) 𝑓(−2) = 4
(e)𝑓(−2) = 7
31. Using the definition of one-to-one, that is, if 𝑓 (𝑎 ) = 𝑓(𝑏) then 𝑎 = 𝑏, determine whether the
function 𝑓 (𝑥) =
𝑥+1
𝑥+2
is one-to-one.
32. Find the inverse of each function:
(a)
1
0.9
𝑟
𝑇(𝑟)
2
2.3
3
3.4
4
1
5
0
(b) 𝑆(𝑡) = 𝐴𝑡 3 + 𝐵 where 𝐴 and 𝐵 are nonzero constants
33. Use the functions 𝑝 and 𝑞 below to determine the following:
𝑥
𝑝(𝑥)
-2
-1
0
2
5
2
4
5
9
15
(a) 𝑝 −1 (15)
(b) 𝑞−1 (3)
(d) (𝑞−1 ∘ 𝑝)(−1)
𝑥
𝑞(𝑥)
-2
-1
0
2
5
3
11
4
6
2
(c) (𝑝 −1 + 𝑞−1 )(4)
(e) (𝑞 ∘ 𝑝−1 )(9)
(f) (𝑞−1 ∘ 𝑞)(5)
34. Find the following. If it does not exist, state why.
(a) Given 𝑓 (5) = 2, find f −1 (2) = ________.
(b) Given that the domain of g ( x) is (−2, ∞) and the range of g ( x) is (−∞, 7] , find the domain
and range of g −1 ( x) .
35. If the point (2,6) is on the graph of 𝑓(𝑥), which if the following points MUST be on the graph of
𝑓 −1 (𝑥)?
1
(a) (−2, −6)
(b) �2, 6�
(a) 𝑓(25)
(b) 𝑓 −1 (30)
(b) (6,2)
(b) (2, −6)
(b) None of these
36. Let 𝑝 be the price in dollars of an item and 𝑞 be the number of items sold at that price, where
𝑞 = 𝑓(𝑝). What do the following quantities mean in terms of prices and quantities sold?
37. Simplify
2𝑎 𝑛+1
𝑎 2𝑛−1
38. (a) Given that the point (−1, −3) is on the graph of 𝑓(𝑥) and 𝑓 (−2) = −9, determine the values of
𝐶 and 𝑏 so that 𝑓(𝑥) = 𝐶 ∙ 𝑏 𝑥 .
(b) Sketch the graph of 𝑓 and label any intercepts and asymptotes.
(c) On what intervals is 𝑓 increasing?
(d) On what intervals is 𝑓 decreasing?
39. The groundskeeper of a local high school estimates that due to heavy usage by the baseball and
softball teams, the pitcher’s mound loses one-fifth of its height every month.
(a) Determine the height of the mound after 3 months if it began at a height of 25 cm.
(b) Determine how long until the pitcher’s mound is less than 16 cm high.
40. Graph the following functions. Label all intercepts and asymptotes. In each case, what is the domain
and range?
(a) 𝑦 = 3𝑥
(b) 𝑦 = 3−𝑥 − 2
(c) 𝑦 = −3𝑥+1
(d) 𝑦 = 3𝑥+1
41. Sketch the graph the following functions. Label all intercepts and asymptotes. In each case, what is
the domain and range?
(a) 𝑓 (𝑥) = log 3 (𝑥 − 2) + 3
(b) 𝑔(𝑥) = − log 4 (𝑥 + 2)
42. If f ( x) =
−32− x + 9 , determine the domain and range of the inverse function f −1 ( x) .
42. Simplify the following.
(a) log 7 147 − log 7 12 + 2 log 7 2
(b) log125 1 + log e
1
− log 4 16
e3
1
43. Determine the domain of 𝑓(𝑥) = log 5 �− 𝑥 �. Solve 𝑓 (𝑥) = 5 for 𝑥.
44. Determine the x - and y -intercepts of the function g ( x) =
− log 2 ( x + 8) + 1 exactly.
45. Determine the domain of the following functions:
(a) 𝑓 (𝑥 ) =
2
√𝑥 2−3𝑥−4
(b) 𝑔(𝑥 ) = log(3𝑥 2 + 7𝑥 + 2)
46. Find the inverse for each of the following functions.
(a) k (=
x ) e5− x − 3
(c) p ( x) =
x
3
2 + 3x
(c) 𝑔(𝑥) = log �
(b) g (=
x) log 2 ( x + 4) + 7
2𝑥 2 +8𝑥
𝑥 2 −2
�
47. Solve the following equations:
(a) log 2 (3 − 𝑥) + log 2 (−𝑥) = 2
(b) log 5 (𝑥 2 − 1) = 3
(c) ln(2 − 4𝑦) − ln(𝑦) = ln(𝑦 − 3)
48. Solve each of the following exponential equations:
(a) 2
−2 x
 1 
= 
 32 
x −3
(b) (𝑒 2𝑥−4 )3 =
𝑒 𝑥+5
𝑒2
(d) 83𝑥−7 = 1
(e)
175
= 50
1 + 5e3 x
(g) 3220 = 42𝑥+6
(h)
e 2 x − e x − 11 =
1
(c) 4(1.05)2𝑡 = 14
(f) 59𝑥 = 47𝑥−3
49. Solve for T in each equation.
(a)
=
A
(
)
P kT
e −1
k
  T 

  3 
(b) P = ln  ln 
50. New employees are given an initial exam and then retested monthly with an equivalent exam. The
average score for employees can be expressed as
S (t ) =
76 − 6·log 3 (t + 1)
where t is measured in months since the initial exam.
(a) What was the average score on the initial exam?
(b) What was the average score on the exam given 26 months after the initial exam?
(c) When would the average score be 54?
51. A population of ladybugs grows according to the limited growth model
=
A 400 − 400e −0.04t
where t is measured in weeks, t ≥ 1 .
(a) How many ladybugs will there be in 20 weeks?
(b) What happens to the population as t grows very large?
(c) When will the population be approximately 300 ladybugs?
52. A valuable painting was purchased in 1980 for $125,000. The painting is expected to double in value
in 15 years. Its value (in thousands of dollars) is modeled by the function 𝑉(𝑡) = 𝑉0 𝑒 𝑘𝑡 , where 𝑡 is
the number of years since 1980. Leave your answers in exact form.
(a) Determine the value of the annual rate 𝑘.
(b) According to this model, when will the value of the painting be $625,000
53. A population of bacteria doubles every generation. Find an equation modeling the growth rate and
determine how many generations it would take to reach over 500 bacteria.
54. Suppose you have $10,000 to invest. Your bank offers you 4.8% compounded quarterly. How much
would you have after 3 years? Simplify as much as possible.
55. The number of bacteria N (t ) in a culture is given by the model N (t ) = 10e kt , where t is the time in
hours. If the culture contains 60 bacteria when t = 4 hours, determine the following.
(a) Determine the rate, k , of growth or decay. Leave answer in exact form.
(b) After how many hours will the culture triple its initial population? Write your answer in exact
form and then round to two decimal places.
56. The half-life of radioactive isotope Carbon-14 is 5730 years. If you have 2 grams remaining after
1000 years, then what was the mass of the initial sample?
57. Suppose a particular radioactive substance has a half-life of 11 years. The initial sample contains
1280g.
(a) Find a function m(t ) that models the amount of the substance left after t years.
(b) Use the model to determine how long it would take the initial sample to decay to 10g. Simplify
completely. You must use the model to find your answer.