MAC 1147 Exam #1a Answer Key
Answer Key
Name:
ID#
Summer 2012
HONOR CODE: On my honor, I have neither given nor received any aid on this
examination.
Signature:
Instructions: Do all scratch work on the test itself. Make sure your final answers
are clearly labelled. Be sure to simplify all answers whenever possible. SHOW ALL
WORK ON THIS EXAM IN ORDER TO RECEIVE FULL CREDIT!!!
No.
1
2
3
4
5
6
7
8
9
10
Score
/4
/6
/8
/2
/2
/3
/2
/2
/2
/2
11
12
13
14
15
16
17
18
19
20
/2
/2
/6
/14
/8
/8
/25
/2
/4
/8
21
22
23
24
25
26
27
28
29
30
Total
/2
/2
/2
/10
/4
/4
/2
/2
/2
/8
/100
(1) Determine whether the relation represents a function. If it is a function, state the
domain and range. (2 points each)
(i) Determine whether the relation represents a function. If it is a function,
state the domain and range.
{(−4, 2), (−1, 1), (0, 0), (−1, −1), (−4, −2)}
(a) not a function
(b) function
domain: {−4, −1, 0}
range: {2, 1, 0, −1, −2}
(c) function
domain: {2, 1, 0, −1, −2}
range: {−4, −1, 0}
(ii) Determine whether the equation |x| + y = 6 defines y as a function of x.
(a) No
(b) Yes
(2) Find the value of the function. (2 points each)
(i) Find f (−1) when f (x) =
(a) − 41
(b) − 45
x2 −4
.
x−3
(c)
3
4
(d) − 25
(ii) Find f (−x) when f (x) = −2x2 − 3x + 4.
(a) 2x2 + 3x + 4
(c) −2x2 + 3x − 4
(b) 2x2 + 3x − 4
(d) −2x2 + 3x + 4
(iii) Find f (x − 1) when f (x) = 5x2 − 3x + 4.
(a) 5x2 − 13x + 12
(c) −13x2 + 5x + 12
(b) 5x2 + 17x + 6
(d) 5x2 − 13x + 6
(3) Find the domain of the function. (2 points each)
(i) f (x) = 4x2 − 16
(a) [2, ∞)
(b) (2, ∞)
(d) (−∞, −2) ∪ (−2, 2) ∪ (2, ∞)
(ii) f (x) =
(c) (−∞, 2)
(e) (−∞, ∞)
3x
x2 +9
(a) (−∞, −3) ∪ (−3, 3) ∪ (3, ∞)
(c) (−∞, 3) ∪ (3, ∞)
(b) (−∞, −3] ∪ [−3, 3] ∪ [3, ∞)
(d) x = 3, −3
(e) (−∞, ∞)
(a) (−∞, ∞)
(b) (−∞, 5)
(c) (−∞, 5]
(d) (5, ∞)
(e) [5, ∞)
(iii) f (x) =
(iv) f (x) =
√
25 − 5x
√9
x−3
(a) (−∞, ∞)
(d) (−∞, 3)
(b) (3, ∞)
(e) (−∞, 3]
(c) [3, ∞)
Use the following graph to answer questions 4 - 13.
(4) Does this graph represent a function? (2 points)
(a) Yes
(b) No
(5) What is the domain and range of the graph? (2 points)
(a) domain: (−∞, 4)
range: (−6, 6)
(b) domain: (−6, 6)
range: (−∞, 4)
(c) domain: (−∞, 4]
range: [−6, 6]
(d) domain: [−6, 6]
range: (−∞, 4]
Domain = (−∞, 4]; Range = (−∞, 6]
(6) What are the x- and y-intercepts of the graph? Note: Not multiple choice. (3
points)
x-intercepts: (−4, 0), (−2, 0), (0, 0), (2, 0), (4, 0)
y-intercept: (0, 0)
(7) What type of symmetry does the graph have? (2 points)
(a) x-axis
(b) even function (c) odd function (d) no symmetry
(8) What is the value of f (1)? (2 points)
(a) 0
(b) 1
(d) none of the above
(c) 6
(9) Is f (−7) greater than −2 or less than −2? (2 points)
(a) f (−7) > −2
(b) f (−7) < −2
(10) For what numbers x is f (x) > 0? (2 points)
(a) [−4, −2] ∪ [4, 5]
(d) (−4, −2) ∪ (0, 2)
(b) (−∞, 4]
(c) (−∞, −4]∪[−2, 0]∪[2, 4]
(e) (−∞, −4) ∪ (−2, 0) ∪ (2, 4)
(11) For what numbers x is f (x) ≤ 0? (2 points)
(a) [−4, −2] ∪ [4, 5]
(b) (−∞, 4]
(d) (−4, −2) ∪ (0, 2)
(e) (−∞, −4) ∪ (−2, 0) ∪ (2, 4)
(c) (−∞, −4]∪[−2, 0]∪[2, 4]
(12) On what intervals is the function increasing and decreasing? (2 points)
(a) increasing: (−∞, −∞) ∪ (−1, −6) ∪ (3, −4)
decreasing: (4, −6) ∪ (6, −4)
(b) increasing: (4, −6) ∪ (6, −4)
decreasing: (−∞, ∞) ∪ (−1, −6) ∪ (3, −4)
(c) increasing: (−∞, −3) ∪ (−1, 1) ∪ (3, 4)
decreasing: (−3, −1) ∪ (1, 3)
(d) increasing: (−3, −1) ∪ (1, 3)
decreasing: (−∞, −3) ∪ (−1, 1) ∪ (3, 4)
(13) State whether each point is a local maximum, local minimum, or neither. (1 point
each)
(i) (−4, 0)
(a) local maximum
(b) local minimum
(c) neither
(b) local minimum
(c) neither
(b) local minimum
(c) neither
(b) local minimum
(c) neither
(b) local minimum
(c) neither
(b) local minimum
(c) neither
(ii) (−3, 4)
(a) local maximum
(iii) (−1, −6)
(a) local maximum
(iv) (1, 6)
(a) local maximum
(v) (3, −4)
(a) local maximum
(vi) (4, 0)
(a) local maximum
(14) Let f (x) be the piecewise-defined function
(
−x3 − 1 if − 2 ≤ x < 0
f (x) = √
x
if 0 ≤ x ≤ 4
(i) Find f (4). (2 points)
2
(ii) Sketch the graph of f (x). (8 points)
(iii) State the domain and range of f (x). (4 points)
Domain = [−2, 4]; Range = (−1, 7]
(15) Use the graph of the function f illustrated below to graph the function H(x) =
2f (−x) − 3. (8 points)
(16) For the function below, state what types of transformations have been done to
the parent function to arrive at the given function. Then graph the function. Be
sure to state what order the transformations are done in. (8 points)
1 f (x) = − x − 3
2
1st : horizontal stretch
2nd : reflection about x-axis
3rd : down 3
(17) Let f (x) be the quadratic function given below. Answer the following questions
about f (x).
f (x) = −2x2 − 4x + 6
(i) Does the graph of f open up or down? How do you know? (2 points)
Down
(ii) What is the vertex (h, k) of f ? (4 points)
(−1, 8)
(iii) What are the intercepts of f ? (4 points)
x-intercepts: (−3, 0), (1, 0); y-intercept: (0, 6)
(iv) What is the domain of f ? (1 point)
(−∞, ∞)
(v) What is the range of f ? (2 point)
(−∞, 8]
(vi) What are the intervals of increase and decrease of f ? (4 points)
Increasing: (−∞, −1); Decreasing: (−1, ∞)
(vii) What does the graph of f look like? (8 points)
(18) For the polynomial function given below, list each real zero and its multiplicity.
Determine whether the graph crosses through the x-axis or touches the x-axis
and turns around at each x-intercept. (2 points)
f (x) = 4(x + 7)(x − 1)4
(a) −7, multiplicity 1, touches the x-axis and turns around; 1, multiplicity 4,
crosses through the x-axis
(b) 7, multiplicity 1, crosses through the x-axis; −1, multiplicity 4, touches the
x-axis and turns
(c) 7, multiplicity 1, touches the x-axis and turns around; −1 multiplicity 4,
crosses through the x-axis
(d) −7, multiplicity 1, crosses through the x-axis; 1, multiplicity 4, touches the
x-axis and turns around
(e) None of the above
(19) For parts (i) and (ii), let
f (x) = −x2 (x + 3)3 (x2 − 1)
(i) For the polynomial function given above, determine the end behavior; that
is, find the power function that the graph of f resembles for large values of
|x|. (2 points)
(a) y = x3
(b) y = x7
(c) y = −x3
(d) y = −x7
(ii) Determine the graph of the power function in part (a). (2 points)
(a)
(c)
(b)
(d)
(20) Graph the polynomial function given below. (8 points)
f (x) = (x − 2)(x + 2)2 (x − 1)2
(21) Find the vertical asymptotes, if any, of the rational function. (2 points)
16 − x2
f (x) = 2
x + 5x + 4
(a) x = −1
(b) x = −1, x = −4
(d) x = −1, x = 4
(c) x = 1, x = −4
(e) None of the above
(22) Find the horizontal or oblique asymptote, if any, of the rational function. (2
points)
−3x2
f (x) = 2
x + 4x − 45
(a) y = −3x + 1
(d) y = −3
(b) y = x − 3
(c) y = 0
(e) None of the above
(23) Find the horizontal or oblique asymptote, if any, of the rational function. (2
points)
2x3 + 11x2 + 5x − 1
f (x) =
x2 + 6x + 5
(a) y = 0
(b) y = 2
(d) y = 2x + 1
(e) None of the above
(c) y = 2x − 1
(24) Graph the function. (Hint: You will need to determine the domain, vertical
asymptotes and/or holes, intercepts, and end behavior.) (10 points)
f (x) =
3x2 (x − 1)(x + 4)3 (x + 2)
(x + 1)4 (x − 2)2 (x + 2)
(25) Solve each inequality algebraically. (2 points each)
(i) x3 ≥ 6x2
(a) (0, 6]
(b) (−∞, 0] ∪ [6, ∞)
(c) (−∞, 6]
(e) None of the above
(d) [6, ∞)
(ii) (x + 2)2 (x + 4) < 0
(a) (−∞, −4)
(d) (−4, −2) ∪ (−2, ∞)
(b) (−2, ∞)
(c) (−∞, −4) ∪ (−4, −2)
(e) None of the above
(26) Solve each inequality algebraically. (2 points each)
(i)
x+9
x+5
<2
(c) (−∞, −5) ∪ (5, 8)
(a) (−∞, −5) ∪ (−1, ∞)(b) (−5, 1)
(d) (−∞, −1) ∪ (5, ∞)
(ii)
x2 (x−12)(x+3)
(x−4)(x+8)
(e) None of the above
≥0
(a) (−8, −3] ∪ (4, 12]
(b) (−∞, −8) ∪ [−3, 4) ∪ [12, ∞)
(c) (−∞, −8] ∪ [−3, 4] ∪ [12, ∞)
(d) (−∞, −8) ∪ [−3, 0) ∪ (0, 4) ∪ [12, ∞)
(e) None of the above
(27) Give the maximum number of zeros the polynomial function may have. Use
Descartes’ Rule of Signs to determine how many positive and how many negative
zeros it may have. (2 points)
f (x) = x6 + 4x5 + 4x4 + 3x3 − x2 − 5x + 4
(a) 6; 2 or 0 positive zeros; 5, 3, or 1 negative zeros
(b) 6; 3 or 1 positive zeros; 4, 2, or 0 negative zeros
(c) 6; 4, 2, or 0 positive zeros; 2 or 0 negative zero
(d) 6; 2 or 0 positive zeros; 4, 2, or 0 negative zero
(e) None of the above
(28) List the potential rational zeros of the polynomial function. (2 points)
f (x) = −2x3 + 3x2 − 2x + 8
(a) ± 41 , ± 12 , ±1, ±2, ±4, ±8
(d) ± 18 , ± 14 , ± 12 , ±1, ±2, ±4, ±8
(b) ± 21 , ±1, ±2, ±4
(e) ± 21 , ±1, ±2, ±4, ±8
(29) Use the given zero to find the remaining zeros of the function. (2 points)
f (x) = x4 − 45x2 − 196; zero: − 2i
(a) 2i, 14, −14
(d) 2i, 14i, −14i
(b) 2i, 7, −7
(c) 2i, 7i, −7i
(e) None of the above
(30) Find the real and complex zeros of the polynomial function and write f in factored
form. (8 points)
f (x) = x4 − x3 − x2 − x − 2
zeros: x = 2, −1, i, −i
f (x) = (x − 2)(x + 1)(x − i)(x + i)
Extra Blank Graphs.