ATM Final Review: Polynomial and non-Polynomial functions and their graphs
Name__________________________________
Teacher: __________________________Pd: __________
Students are to answer the following and show ALL work either on their paper, or on a separate sheet of paper. All answers
need to be either circled or highlighted.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Factor by grouping. Assume any variable exponents represent whole numbers.
1) x3 + 8x - 4x2 - 32
A) (x + 4)(x2 + 8)
B) (x - 4)(x2 - 8)
C) (x - 4)(x2 + 8)
D) (x - 4)(x + 8)
Factor the trinomial, or state that the trinomial is prime.
2) x2 - 2x - 24
A) (x + 6)(x + 4)
B) (x + 6)(x + 1)
C) (x - 6)(x + 4)
D) prime
C) (7x + 2)(x - 5)
D) 7(x - 5)(x - 2)
3) 7x2 - 19x + 10
A) (7x - 5)(x - 2)
B) (7x - 5)(7x + 2)
4) 6x2 + 13xy + 6y2
A) (6x + 3y)(x + 2y)
C) (2x + 3y)(3x + 2y)
D) prime
6) (81x4 - 16)
A) (3x + 2)2 (3x - 2)2
C) (9x2 + 4)(9x2 - 4)
8) 16x2 + 8x + 1
A) (x + 4)2
3)
5)
B) (x2 - 1)(x2 - 1)
C) (x2 + 1)(x2 + 1)
2)
4)
B) (2x - 3y)(3x - 2y)
D) prime
Factor the difference of two squares.
5) x4 - 1
A) (x2 + 1)(x + 1)(x - 1)
Factor the perfect square trinomial.
7) x2 - 20x + 400
A) (x - 20)2
1)
B) (9x2 + 4)(3x + 2)(3x - 2)
D) (9x2 + 4)(9x2 + 4)
B) (x + 20)(x - 20)
C) (x + 20)2
B) (4x + 1)(4x - 1)
C) (4x + 1)2
7)
D) prime
8)
D) prime
Factor using the formula for the sum or difference of two cubes.
9) 27x3 + 8
A) (3x - 2)(9x2 + 6x + 4)
B) (3x + 2)(9x2 + 6x + 4)
C) (3x + 2)(9x2 - 6x + 4)
D) (3x + 2)(9x2 + 4)
10) 125x3 - 8
A) (5x + 2)(25x2 - 10x + 4)
C) (5x - 2)(25x2 - 10x + 4)
B) (5x - 2)(25x2 + 4)
D) (5x - 2)(25x2 + 10x + 4)
1
6)
9)
10)
Factor completely, or state that the polynomial is prime.
11) 12x3 - 192x
11)
A) x(x + 4)(12x - 48)
C) 12(x + 4)(x2 - 4x)
B) 12x(x + 4)(x - 4)
D) prime
12) x3 - 6x 2 - 16x + 96
A) (x - 6)(x + 4)(x - 4)
B) (x - 6)(x - 4)2
C) (x + 6)(x + 4)(x - 4)
D) prime
13) x2 + 64
A) (x - 8)2
13)
B) (x + 8)2
C) (x + 8)(x - 8)
14) 6x3 - 6
A) 6(x + 1)(x2 - x + 1)
C) 6(x3 - 1)
2 4
,
B)
3 3
B) 1, -
9
4
D) {1, 6}
2
4
C) - , 3
3
2
1
D) - , 9
2
C) 1
15)
16)
D) {-4, 1}
18)
-8 - 2 -8 + 2
,
2
2
19) x2 + 10x + 34 = 0
A) {-5 + 3i, -5 - 3i}
C) {-2, -3}
17)
18) 2x2 + 8x + 7 = 0
-4 - 2 -4 + 2
,
A)
2
2
C)
14)
D) prime
17) 7 - 7x = (4x + 9)(x - 1)
A) {-1, 4}
D) prime
B) 6(x - 1)(x2 + x + 1)
Solve the following quadratic equations.
15) x2 = x + 6
A) {2, 3}
B) {-2, 3}
16) 9x2 + 18x + 8 = 0
2
4
,A)
3
3
12)
B) {-5 - 9i, -5 + 9i}
B)
-4 - 30 -4 + 30
,
2
2
D)
-4 - 2 -4 + 2
,
4
4
C) {-2, -8}
2
D) {-5 + 3i}
19)
Use the graph to determine the x- and y-intercepts.
20)
20)
A) x-intercept: -1; y-intercept: -8
C) x-intercept: -1; y-intercept: 8
B) x-intercept: 1; y-intercept: 8
D) x-intercept: -8; y-intercept: 8
21)
21)
A) x-intercepts: -7, 7; y-intercept: -7
C) x-intercepts: -7, 7; y-intercept: 0
B) x-intercepts: -7, 7
D) y-intercept: -7
3
22)
22)
A) x-intercept: 8; y-intercepts: -4, 2
C) x-intercepts: -4, 2; y-intercept: 8
B) x-intercept: -4; y-intercepts: 2, 8
D) x-intercept: 2; y-intercept: 8
23)
23)
A) y-intercept: 4
B) x-intercept: 4
C) x-intercept: -4
4
D) y-intercept: -4
24)
24)
A) x-intercept: 3; y-intercepts: -3, 1
C) x-intercept: -3; y-intercepts: 1, 3
B) x-intercept: 1; y-intercept: 3
D) x-intercepts: -3, 1; y-intercept: 3
25)
25)
A) x-intercept: -2; y-intercepts: 2, 1, 5
C) x-intercepts: 2, 1, 5; y-intercept: -2
B) x-intercepts: -2, 1, -5; y-intercept: -2
D) x-intercept: -2; y-intercepts: -2, 1, -5
5
26)
26)
A) x-intercepts: -8, 8
C) y-intercepts: -7, 7
B) x-intercepts: -8, 8; y-intercepts: -7, 7
D) x-intercepts: -7, 7; y-intercepts: -8, 8
Match the story with the correct figure.
27) Mark started out by walking up a hill for 5 minutes. For the next 5 minutes he walked down a steep
hill to an elevation lower than his starting point. For the next 10 minutes he walked on level
ground. For the next 10 minutes he walked uphill. Determine which graph of elevation above sea
level versus time illustrates the story.
A)
B)
C)
D)
6
27)
Find the distance between the pair of points.
28) (-5 5, 3) and (-3 5, 7)
A) 6
B) 36
C) 18
29) (-4 17, -2) and (-3 17, 6)
81
A)
B) 81
2
D) 5
29)
C) 8
Find the midpoint of the line segment whose end points are given.
30) (9 3, 5 7) and (14 3, 10 7)
-5 3 -5 7
23 3 15 7
5 3 5 7
,
)
,
)
,
)
A) (
B) (
C) (
2
2
2
2
2
2
31) (6 6, 7 2) and (11 6, 10 2)
-5 6 -3 2
5 6 3 2
,
)
,
)
A) (
B) (
2
2
2
2
D) 9
30)
D) (23 3, 15 7)
31)
C) (
17 6 17 2
,
)
2
2
D) (17 6, 17 2)
Complete the square and write the equation in standard form. Then give the center and radius of the circle.
32) x2 + y2 - 16x - 8y + 80 = 49
A) (x - 8)2 + (y - 4)2 = 49
B) (x - 8)2 + (y - 4)2 = 49
(8, 4), r = 7
32)
(-8, -4), r = 49
C) (x - 4)2 + (y - 8)2 = 49
(4, 8), r = 7
D) (x - 4)2 + (y - 8)2 = 49
(-4, -8), r = 49
33) 6x2 + 6y2 = 36
A) x2 + y2 = 6
(0, 0), r = 6
C) (x - 6)2 +(y - 6)2 = 6
(6, 6), r =
28)
B) x2 + y2 = 6
(0, 0), r = 6
D) x2 + y2 = 36
33)
(0, 0), r = 6
6
Write the standard form of the equation of the circle with the given center and radius.
34) (-9, 7); 10
A) (x + 7)2 + (y - 9)2 = 10
B) (x + 9)2 + (y - 7)2 = 100
2
2
C) (x - 9) + (y + 7) = 100
D) (x - 7)2 + (y + 9)2 = 10
35) (0, 10); 15
A) (x + 10)2 + y2 = 225
C) (x - 10)2 + y2 = 225
B) x2 + (y + 10)2 = 15
D) x2 + (y - 10)2 = 15
7
34)
35)
Graph the equation.
36) (x - 5)2 + (y - 4)2 = 4
36)
A)
B)
Domain = (3, 7), Range = (2, 6)
Domain = (-7, -3), Range = (-6, -2)
8
Graph the equation and state its domain and range. Use interval notation
37) x2 + y2 = 25
A)
37)
B)
Domain = (-5, 5); Range = (-5, 5)
Domain = (- 5,
5); Range = (- 5,
The graph of a function f is given. Use the graph to answer the question.
38) Find the numbers, if any, at which f has a relative minimum. What are the relative minima?
A) f has a relative minimum at x = -3; the relative minimum is 0
B) f has a relative minimum at x = -3 and 3; the relative minimum is 0
C) f has a relative minimum at x = 0; the relative minimum is 1
D) f has no relative minimum
9
5)
38)
Use the graph of the given function to find any relative maxima and relative minima.
39) f(x) = x3 - 3x2 + 1
A) no maximum or minimum
C) maximum: (0, 1); minimum: none
39)
B) maximum: none; minimum: (2, -3)
D) maximum: (0, 1); minimum: (2, -3)
Use the Leading Coefficient Test to determine the end behavior of the polynomial function. Then use this end behavior
to match the function with its graph.
40) f(x) = 4x2 - 2x + 2
40)
A) rises to the left and falls to the right
B) falls to the left and falls to the right
C) falls to the left and rises to the right
D) rises to the left and rises to the right
10
Solve the problem.
41) A herd of deer is introduced to a wildlife refuge. The number of deer, N(t), after t years is
described by the polynomial function N(t) = -t4 + 24t + 100. Use the Leading Coefficient Test to
41)
determine the graph's end behavior. What does this mean about what will eventually happen to
the deer population?
A) The deer population in the refuge will grow out of control.
B) The deer population in the refuge will die out.
C) The deer population in the refuge will be displaced by "oil" wells.
D) The deer population in the refuge will reach a constant amount greater than 0.
42) The following table shows the number of DWI arrests in a county for the years 1994-1998, where 1
represents 1994, 2 represents 1995, and so on.
Year, x DWI arrests, T
1994, 1
4216.1
1995, 2
4258.56
1996, 3
4304.6
1997, 4
4342.44
1998, 5
4394.3
This data can be approximated using the third-degree polynomial
T(x) = -0.63x3 + 0.57x2 + 57.16x + 4159.
Use the Leading Coefficient Test to determine the end behavior to the right for the graph of T. Will
this function be useful in modeling the number of DWI arrests over an extended period of time?
Explain your answer.
A) The graph of T decreases without bound to the right. Since the number of larceny thefts will
eventually decrease, the function T will be useful in modeling the number of DWI arrests
over an extended period of time.
B) The graph of T decreases without bound to the right. This means that as x increases, the
values of T will become more and more negative and the function will no longer model the
number of DWI arrests.
C) The graph of T approaches zero for large values of x. This means that T will not be useful in
modeling the number of DWI arrests over an extended period.
D) The graph of T increases without bound to the right. This means that as x increases, the values
of T will become large and positive and, since the values of T will become so large, the
function will no longer model the number of DWI arrests.
11
42)
Use the Leading Coefficient Test to determine the end behavior of the polynomial function. Then use this end behavior
to match the function with its graph.
43) f(x) = -2x2 - 3x - 1
43)
A) falls to the left and falls to the right
B) falls to the left and rises to the right
C) rises to the left and rises to the right
D) rises to the left and falls to the right
12
44) f(x) = 6x3 - 3x2 - 3x - 1
A) rises to the left and rises to the right
B) rises to the left and falls to the right
C) falls to the left and rises to the right
D) falls to the left and falls to the right
13
44)
45) f(x) = -8x3 - 2x2 + 4x + 2
A) rises to the left and rises to the right
B) rises to the left and falls to the right
C) falls to the left and rises to the right
D) falls to the left and falls to the right
14
45)