A2A MAT115 Practice Exam 4 Answers: Be aware that the following problems are not inclusive but just examples of
the content listed above. Go over the homework sets for further examples. Correct the problems on the hw that you got
wrong. The following problems are designed to match the content above.
 b  b 2  4ac
Quadratic Formula:
2a
Vertex:
b
2a
1. The graph of f(x) is given in the following figure. Choose the appropriate answer from each list or fill in the blank.
When writing in roots enter from smallest to largest values and include repeated roots in your answer
a. The domain of f(x) = (, )
b. The range of f(x) = (, )
c. Estimate the local minima as an order pair (A,B). Round to
the nearest 0.5 (2,  2.5)
d. Estimate the local maxima as an ordered pair (A,B). Round
to the nearest 0.5 (1.5, 8.5)
e. On what intervals is f(x) always positive (3, 1)  (3, )
f.
On what intervals is f(x) always negative (,  3)  (1, 3)
G(x)
2. For the graph of G(x) given to the right (round to the nearest integer):
a. Find the local minima and maxima for each graph
Local Min: (-1, -3)
Local Max: (-3, 0), (2, 4)
b. Find the x and y-intercepts for each graph
x-int: -3, 1/2, 3
.y-int: -1
c. Find the multiplicity of the x-intercepts for each graph
The three zero’s are x= -3, Mult=2; x=1/2, Mult=1; x = 3, Mult=1 (touch, cross, cross)
d. Find the domain and range in interval notation.
Domain:
 , 
Range:
 , 4
e. Determine if the degree of the graphs are odd or even The graph is of even degree
f.
Is the graph is positive or negative? negative
g. What are the intervals of increase and decrease?
Intervals of increase:  ,  3  (1, 2)
Intervals of decrease  3,  1  (2, )
h. What is the power function for each graph?  ax 4 where a = 2/27
i.
Represent each graph in factored form of the lowest degree. (Don’t forget to include the sign, greatest
common factor, and include multiplicity.) G x   
2
x  32 2 x  1x  3
27
For the graph of H(x) given to the right (round to the nearest integer):
a. Find the local minima and maxima for each graph
Local Min: (2, -2)
H(x)
Local Max: (-2, 2)
b. Find the x and y-intercepts for each graph
x-int: -4, 0, 4
y-int: 0
c. Find the multiplicity of the x-intercepts for each graph
The three zero’s are x= -4, Mult=1; x=0, Mult=1; x = 4, Mult=1 (cross, cross, cross)
d. Find the domain and range in interval notation.
Domain:
 , 
Range:
 , 4
e. Determine if the degree of the graphs are odd or even The graph is of odd degree
f.
Is the graph is positive or negative? postive
g. What are the intervals of increase and decrease?
Intervals of increase:  ,  2  (2, )
Intervals of decrease  2, 2
h. What is the power function for each graph? x 3
i.
Represent each graph in factored form of the lowest degree. (Don’t forget to include the sign, greatest
common factor, and include multiplicity.) H x  x x  4x  4
3. Consider the quadratic function f ( x)   x 2  x  2
a. Find the vertex of the parabola by completing the square and using the vertex formula.
2
1
9

1 9
p  x    x   
 , 
2
4

2 4
b. Is the vertex a maximum or minimum? Maximum
4. Given g ( x)  3x 2  6 x  10 , find the average rate of change between the points x = –2 and x = 3.
The average rate of change is 9
5. Given the function f ( x)  2 x 2  4 x  3 , find the x- and y-intercepts, vertex, the axis of symmetry, concavity,
domain, and range. Graph the function using what you have found.
a. The x-intercepts are
 2  10
= (.58, 0) and (-2.58, 0).
2
b. The y-intercept = (0, 3).
c. The vertex is (-1, 5).
d. The axis of symmetry is x = -1.
e.
f.
g.
The graph has a negative in the front, so it is concave down.
y-intercept
x-intercept
x-intercept
The domain is  ,  .
The range is  ,5 .

6. Mike wants to enclose a rectangular play area for his rabbits next to his large barn using 30 feet of fencing.
What dimensions will maximize the fenced area if the barn is used for one side of the play area?
7.5 feet by 15 feet
7. Mona Kalini gives a walking tour of Honolulu to one person for $49. To increase her business, she advertised at the
National Orthodontist Convention that she would lower the price by $1 per person for each additional person up to 49
people. Mona’s revenue from her revenue can be modeled as a function of the number of people on the tour:
R( x)  49 x  x 2 . How many people on the tour would maximize her revenue? What is the maximum revenue for
her tour? Round to 25 people for maximum people on the tour.
Max revenue is $600
8. An object is shot upward from the floor of a canyon 160 feet deep with an initial velocity of 120 ft/sec. The height of
the object can be modeled by the function h(t )  16t 2  120t 160 .
a. What is the maximum height the object reaches? t = 3.75 sec  max ht at 65 feet
b. How long will it take the object to return to the ground? 5.765 sec
c. How long will it take for the object rise to the height of the canyon wall? Never gets
that high. The max ht of the vertex is 65 feet so does not reach 106 feet.
9.
a. Find the average rate of change for the tangent line on the graph to the right. 2/3
b. What is another way of describing the average rate of change of the tangent line? Slope
of the tangent line.
c. Find the equation of the tangent line. y = ½ x – 13/3
10. Given the functions find the following if they exist: x- and y-intercepts, vertical and horizontal asymptotes, domain
and range (in set notation), and graph the function using this information.
a. f  x  
x2
x2  9


y-intercept:  0,
x-intercepts: (– 2, 0)
Vertical Asymptotes:
x  3

Horizontal/Slant Asymptotes: y = 0.

Domain is x x  , x  3
b.
g ( x) 
x-intercepts:
2x
2
2

9
Range:
y
y  
 3x  1
x2
3  17
= –.28, 1.78
4
Horizontal/Slant Asymptotes:
x
½
2
Vertical Asymptotes: x =
The domain is.
y-intercept:
2x+ 1
x  , x  2
11. Which of the following factors are roots or zeros of the polynomial P( x)  4 x 4  6 x 3  18 x 2  2 x  6
4x  2
x 3
x 1
12. Solve the following inequalities (one is rational and one is quadratic) by finding critical points and using test points.
Write you answer in interval notation.
a.
b.
2
1

x 1 x 1
4 z 2  12 z  9


x3
0 
x  1x  1
4 z 2  12 z  9  0
[-3, -1), 1, 
-3

 ,1.51.5, 
-
+
+
-1
+
1
+
1.5
13. Use synthetic division to divide f ( x)  x3  13x  12 by x  4 . Write your answer in factored form.
-4|
1
0 -13 12
-4 16 -12
1 -4 -3 0
f (x)  ( x  4) ( x 2  4 x  3)
.
14. If the degree of P ( x)  7 and the given zeros are x  12 , x  7 , x  i 10 , x  4  6 , x  2  6i , then
write P (x ) in factored form.
P( x)  ( x  12)( x  7)( x  i 10 )( x  i 10 )( x  4  6 )( x  2  6i )( x  2  6i )
15. Find each of the following for the rational graph given to the right.
a.
b.
c.
d.
e.
Vertical asymptote:
Horizontal asymptote:
x-intercept:
y-intercept:
Domain:
x = -4
y=3
x=–2
y=1½
x | x  , x  4
16. Graph the following functions using transformations. State the domain and range. (Remember you must reflect across the y-axis
before you shift horizontally)
a.
g ( x)  e  x  2  1
d.
k ( x)  1  log 2 ( x  3)
b.
f ( x)  e x 3  4
e.
p( x)  log 2 (2 x  3)
c.
k ( x)  log 2 ( x  2)  1
17. Write each logarithmic equation in exponential form each exponential equation in logarithmic form.
a.
3  log 2 8
1
 log 81 9
b.
2
1
23  8
81 2  9
1
c. 3 
9
1
log 3  2
9
2
d. e
3
4
 2.117
ln( 2.117) 
3
4
18. Determine the value of each logarithm without using a calculator.
a. log 11 121
=2
b. log 81 9 = 1/2
c. log 7 1
49
= –2