Pre Cal Practice Questions for Test 1
Content
Practice Questions Test 1
−1/2
144x2 y −3
Write your answer using positive exponents and sim16x−4 y 8
plified roots where appropriate.
1) Simplify
2) Simplify
x2 − 16
x3 + 512
· 2
2
x + 4x x + 4x − 32
3) Factor x3 + 6x2 + 6x + 1 completely.
4) If f (x) = (g(2x))2 + 1 and g(x) = 4x − 2 find f (1).
5) Simplify
4x
2
−
6x + 3 3x − 2
6) Solve 3x2 + 7x − 2 = 0
7) Find the center and radius of the circle x2 + y 2 + 4x − 10y + 4 = 0
8) Find an equation for the circle that has the points (2, 4) and (8, 12) as endpoints of one of its diameters.
9) A certain circle is centered at the origin of the cartesian plane, and the point
(5, 12) lies on this circle. Find an equation for this circle.
10) Find the vertex, x− intercepts, y intercept, and the equation of the axis of
symmetry of the parabola y = x2 + 8x + 15.
11) What is the domain of f (x) = x2 + 8x + 15? Find the coordinates of any
local maxima or minima of this function. Is there an absolute maximum? Absolute
Minimum? On what intervals is f increasing? Decreasing?
12) Is the relation |y| = 1 + 4x is a function? Why or why not? Justify any
claims.
13) Consider the ellipse x2 + 49y 2 = 1. What are the coordinates of its center?
How long is the major axis? How long is the minor axis? Does the major axis run
along the x axis or y axis? Sketch the ellipse. Find the coordinates of the foci of
this ellipse.
14) What is the average rate of change of f (x) =
√
x on the interval [4, 25]?
15) Find the difference quotient for f (x) = x3 .
16) Find the domain of f (x) =
√
8 − 2x +
val notation.
1
1
. Write your answer in interx−1
2
17) Find the equations of the asymptotes of the hyperbola 4x2 − 9y 2 = 144.
18) A hotel clerk finds that typically when he charges 88 dollars per room per
night, he fills 50 of his rooms with customers. For each 2 dollar increase in charge,
he fills one less room, and for each 2 dollar decrease in charge he fills one more
room. (For instance, when he charges 90 dollars he normally fills only 49 rooms, 92
dollars normally 48 rooms, and 86 dollars normally 51 rooms, etc). What should
he charge customers if he wishes to maximize the total amount of money he brings
in for the night?
19) Suppose f (1) = 4, f (2) = 3, f (3) = 1, f (4) = 2, g(1) = 3, g(2) = 2, g(3) = 4,
g(4)
= 1,h(1) = 1, h(2) = 2, h(3) = 4, and h(4) = 3. Determine the value of
(h(1))
f fg(f
(4))
20) Use a graphing calculator to graph f (x) = x4 − 29x2 + 100. Use it to find
the coordinates of any local maxima or minima. What is the absolute maximum
value of f (x)? What is the absolute minimum value of f (x)? On what interval(s)
is f increasing? Decreasing?
−1/2
144x2 y −3
Write your answer using positive exponents and sim16x−4 y 8
plified roots where appropriate.
1) Simplify
2) Simplify
x2 − 16
x3 + 512
· 2
2
x + 4x x + 4x − 32
3) Factor x3 + 6x2 + 6x + 1 completely.
4) If f (x) = (g(2x))2 + 1 and g(x) = 4x − 2 find f (1).
5) Simplify
4x
2
−
6x + 3 3x − 2
6) Solve 3x2 + 7x − 2 = 0
7) Find the center and radius of the circle x2 + y 2 + 4x − 10y + 4 = 0
8) Find an equation for the circle that has the points (2, 4) and (8, 12) as endpoints of one of its diameters.
9) A certain circle is centered at the origin of the cartesian plane, and the point
(5, 12) lies on this circle. Find an equation for this circle.
10) Find the vertex, x− intercepts, y intercept, and the equation of the axis of
symmetry of the parabola y = x2 + 8x + 15.
11) What is the domain of f (x) = x2 + 8x + 15? Find the coordinates of any
local maxima or minima of this function. Is there an absolute maximum? Absolute
Minimum? On what intervals is f increasing? Decreasing?
12) Is the relation |y| = 1 + 4x is a function? Why or why not? Justify any
claims.
13) Consider the ellipse x2 + 49y 2 = 1. What are the coordinates of its center?
How long is the major axis? How long is the minor axis? Does the major axis run
along the x axis or y axis? Sketch the ellipse. Find the coordinates of the foci of
this ellipse.
14) What is the average rate of change of f (x) =
√
x on the interval [4, 25]?
15) Find the difference quotient for f (x) = x3 .
16) Find the domain of f (x) =
√
8 − 2x +
val notation.
1
1
. Write your answer in interx−1
2
17) Find the equations of the asymptotes of the hyperbola 4x2 − 9y 2 = 144.
18) A hotel clerk finds that typically when he charges 88 dollars per room per
night, he fills 50 of his rooms with customers. For each 2 dollar increase in charge,
he fills one less room, and for each 2 dollar decrease in charge he fills one more
room. (For instance, when he charges 90 dollars he normally fills only 49 rooms, 92
dollars normally 48 rooms, and 86 dollars normally 51 rooms, etc). What should
he charge customers if he wishes to maximize the total amount of money he brings
in for the night?
19) Suppose f (1) = 4, f (2) = 3, f (3) = 1, f (4) = 2, g(1) = 3, g(2) = 2, g(3) = 4,
g(4)
= 1,h(1) = 1, h(2) = 2, h(3) = 4, and h(4) = 3. Determine the value of
(h(1))
f fg(f
(4))
20) Use a graphing calculator to graph f (x) = x4 − 29x2 + 100. Use it to find
the coordinates of any local maxima or minima. What is the absolute maximum
value of f (x)? What is the absolute minimum value of f (x)? On what interval(s)
is f increasing? Decreasing?
