Math 121 – Practice Problems for Exam 2
Name: ________________________
1. Which one of the following statements is true? Correct the false statements.
a) The function f ( x)  ( x  1)2  1 has a y-intercept at (0, 1).
b) The function f ( x)  x  2  1 is decreasing over the interval (–2, 0)
c) The domain of the function f ( x)  2 x  1 includes all real values of x.
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d) The function f ( x) 
has an x-intercept at (–3, 0).
x3
2. If f ( x)  x  4 , find: a) f(0); b) f(x – 2); c) 2f(x); d) f(2) – f(–5)
 x 2  2 x  3
3. a) Graph the function f ( x)  
; b) Find f (1) ; find f (2) .
 2 x  3 x  3
4. Sketch the graph of f ( x) 
x  2 . Determine its domain and range.
5. Sketch the graph of f ( x)   x  2  3 . Determine its domain and range.
Identify x- and y-intercepts, if they exist.
6. Describe the transformations of the graph of f ( x)  x 2 that would produce the
graph of g ( x)  ( x  2)2  1 .
7. Find the average rate of change of the function f ( x)  x 2  2 x  1 from – 2 to 1.
8. Solve: 2 x 2  32 x
9. Solve: 1  ( x  1)2  0
10. A rectangle has an area of 432 square inches. The width of the rectangle is ¾
times its length. Find the dimensions of the rectangle.
11. Solve by completing the square: x 2  6 x  3  0 . Leave answers in exact form.
12. Write an equation of the parabola in the form y  a( x  h)2  k that has a vertex
at (2, – 5) and another point (0, 3).
13. An object is dropped from a height of 75 feet. Its height h (in feet) at any time t is
given by h  16t 2  75 , where t is measured in seconds. Find the time required
for the object to fall to a height of 35 feet. [Round answer to the nearest tenth.]
14. Solve: 8 
12 1

t
3
x2
. Label x- and y-intercepts, if they
x5
exist, and locate at least two points on each part of the graph.
15. Graph the rational function y  f ( x) 
16. a) Give an example of an odd function that is everywhere increasing.
b) Give an example of an even function that has a zero of multiplicity 2 at the
point (0, 0).
17. Sketch the graph of p( x)  2 x3  3x 2 . Label the x- and y-intercepts. What is the
domain; and the range of p(x)? Find p(1), find p(– 2). For what values of x does
p(x) = 0?
18. Find the zeros of the function g ( x)  4 x 4  10 x3  6 x 2 . State their multiplicities.
19. Over what interval(s) is the function g ( x)  x 2  2 x  1 increasing?
20. A cyclist leaves Toronto at 10 am riding west. At noon, a car starts along the
same road, driving 3 times as fast as the bike. When will the car catch up with
the bike?
21. Construct a scatter diagram for these data points: (1, 1), (6, –1), (–2, 2), (3, 2),
(–5, 4), (4, 0), (3, 1), (–1, –1), (–3, 3), (–4, 1). Draw a linear approximation of the
data and calculate its equation.
Answers: 1) c 2: a) 4; b) x  2 ; c) 2 x  4 ; d) 5. 3) a)The parabola opens downward,
with vertex at (0, 2). The line and the parabola intersect at (1, 1) ; b)
f (1)  1; f (2)  2 . 4) V: (2, 0); other points: (3, 1), (6, 2), (11, 3); D: [2, ∞); R: [0,
∞). 5) V: (2, 3); other points: (3, 2), (6, 1), (11, 0). D: [2, ∞); R: (–∞, 3); x-int: (11, 0);
no y-intercept. 6) Move the vertex of the parabola 2 units to the right, flip it upside down,
and move it 1 unit up. 7)  3 8) x = 0, x = 16. 9) x = 0, x = −2. 10) 18 in. by 24 in.
11) 3  12 [ 3  2 3 ]. 12) y  2( x  2)2  5 . 13) The time required is 1.6 seconds.
14) t  36
. 15) VA: x = −5; HA: y = 1; Pts: (0,  52 ) , (2, 0), (1,  16 ) , (−4, −6), (−6, 8).
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16) Exampls: a) f ( x)  3 x ; b) f ( x)  x 2 . 17) Intercepts: (0, 0) and ( 32 , 0) ; D: (−∞, ∞),
R: (−∞, ∞); p(1) = −1, p(– 2) = −28; x = 0 and x  32 . 18)  3 (1), 0 (2), 12 (1) . 19) The
function is increasing over the interval (1, ∞). . 20) The car will catch up with the bike
at 1 pm. 21) y   14 x  32 is an approximation.