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COLLEGE ALGEBRA, EXAM TWO
There are eight partial credit problems on this exam. Show
all your work on the page on
which the question appears.
Problem 1 (10 points). Find a linear function f(x) such
that the graph of y = f(x) passes
through the points (1,2) and (4, 4) in the xy coordinate plane .
Solution . Write f(x) = mx + b. Set x_{1} = 1, y_{1} = 2, x_{2} = 4, and y_{2} = 4.
Use the slope formula to find m.
So f(x) = 2x + b. Since the graph of y = f(x) passes
through (1,2), we get f(1) = 2.
2(1) + b = 2,
2 + b = 2,
b = 2  2 = 4
Therefore f(x) = 2x  4.
Problem 2 (10 points). Find a linear function f(x) such that f(1) = 2 and the
graph of
y = f(x) is perpendicular to the straight line x  y = 2 in the xycoordinate
plane.
Solution . Write f(x) = mx + b.
(1) Find the slope of the straight line x  y = 2. Solve x  y = 2 for y.
x  y = 2,
y = x  2,
slope = 1
(2) The straight lines y = f(x) and x  y = 2 are perpendicular to each other.
m ۰ 1 = 1,
m = 1
So f(x) = x + b.
(3) Since f(1) = 2, we get (1) + b = 2 or b = 1.
Therefore f(x) = x + 1.
Problem 3 (17 points). Questions (a) through (d) refer to the quadratic function
y = 2x^{2}  4x  2 :
(a) Complete the square for y = 2x^{2}  4x  2.
Solution. In this quadratic function , a = 2, b = 4, and c = 2.
Write
Therefore y = 2(x  1)^{2}  4.
(b) Find the x and y intercepts on the graph of y = 2x^{2}  4x  2.
Solution. (1) Find the xintercepts.
(2) Find the yintercept.
xIntercept(s): and
yIntercept: 2
(c) Graph the function y = 2x^{2}  4x  2. Specify the
vertex and the axis of symmetry.
The axes are marked o® in oneunit intervals.
Vertex : (1,4)
Axis of Symmetry: x = 1
(d) Determine whether the function y = 2x^{2}  4x  2 has
the maximum value or the
minimum value .
Maximum Value
Minimum Value
Problem 4 (10 points). A point P(x, y) lies on the graph of the function y = x^{2}
 1.
Express the distance from P(x, y) to the origin O(0, 0) as a function of x.
Solution. Let d be the distance from P(x, y) to O(0, 0). By the distance formula , we get
Problem 5 (14 points). Suppose that the height of a
baseball thrown straight up is given
by the function
H(t) = 16t^{2} + 64t
where H is in feet and t is in seconds.
(a) Find the maximum height that the baseball can reach.
Solution. H(t) = 16t^{2} + 64t is a quadratic function with a = 16, b = 64, and c
= 0.
Write
Answer: The maximum height that the baseball can reach is 64 feet.
(b) Find the time at which the baseball hits the ground.
[ Hints: The height of the baseball is 0 feet when it hits the ground. ]
Solution. Find the time t when H(t) = 0.
Answer: The baseball hits the ground at the time t = 4 seconds.
Problem 6 (12 points). Questions (a) and (b) refer to the polynomial function
y = (x  1)^{3}  2 :
(a) Find the x and y intercepts on the graph of y = (x  1)^{3}  2.
Solution. (1) Find the xintercept.
(2) Find the yintercept.
xIntercept(s):
yIntercept: 1
(b) Graph the function y = (x  1)^{3}  2. The axes are
marked off in oneunit intervals.
Solution. Steps :
(1) Sketch the graph of y = x^{3}.
(2) Reflect the graph of y = x^{3} about the xaxis to get the graph of y = x^{3}.
(3) Shift the graph of y = x^{3} to the right by 1 unit to get the graph of y =
(x1)^{3}.
(4) Shift the graph of y = (x1)^{3} down by 2 units to get the graph of y =
(x1)^{3}2.
Problem 7 (14 points). Questions (a) through (c) refer to
the polynomial function
y = (x + 2)(x  1)(x  3) :
(a) Determine the x and yintercepts on the graph of y = (x + 2)(x  1)(x  3).
Solution. (1) Find the xintercepts.
(2) Find the yintercept.
xIntercept(s): 2, 1, and 3
yIntercept: 6
(b) Sketch the excluded regions for the graph of y = (x +
2)(x  1)(x  3). The axes are
marked off in oneunit intervals.
Solution. Determine the signs of y = (x + 2)(x  1)(x  3).
The excluded regions are the shaded regions in the following figure.
(c) Sketch the graph of y = (x + 2)(x  1)(x  3). The
axes are marked off in oneunit
intervals.
Solution.
Problem 8 (13 points). Questions (a) through (c) refer to the rational function
(a) Write out the domain of the function
in
interval notation.
Solution. This function is defined when (x  2)(x + 1) ≠ 0.
Domain: (∞,1) ∪ (1, 2) ∪ (2,∞)
(b) Find the x and yintercepts on the graph of
Solution. (1) Find the xintercepts.
(2) Find the yintercept.
xIntercept(s): 2 and 1
yIntercept: 2
(c) Find the horizontal and vertical asymptotes to the graph of
Solution. Use the approximation method to determine the horizontal asymptote.
when x is very large.
Horizontal Asymptote: y = 2
Vertical Asymptote(s): x = 1 and x = 2
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