College Algebra Practice Test3
Name____________________
Problem 1: Sketch the graph of the function
.
Step 1: Reduce the rational function to lowest terms and check for any
open holes in the graph.
*Factor
*Cancel the common factor of x
The factor of x canceled out and there were no factors of x left in the denominator.
This means there is an open hole on the graph at x = 0.
x = 0:
1
*Plug in a 0 for x
There is an open hole at (0, -4/3).
Step 2: Find all of the asymptotes and draw them as dashed lines.
Vertical Asymptote:
So now we want to find where the new denominator is equal to 0:
*Set den. = 0
There is one vertical asymptote: x = -3.
Horizontal Asymptote:
Since the degree of the numerator is one degree higher than the degree of the
denominator, there is a slant asymptote and no horizontal asymptote.
Slant Asymptote:
Applying long division to this problem we get:
2
The answer to the long division would be
.
The equation for the slant asymptote is the quotient part of the answer which
would be
.
Step 3: Determine the symmetry.
3
Since
odd.
, the function is neither even nor
Step 4: Find and plot any intercepts that exist.
x-intercept:
What value are we going to use for y? You are correct if you said 0.
*Plug in 0 for y
*Mult. both side by the LCD (x + 3)
*Factor
*Set 1st factor = 0
*Set 2nd factor = 0
4
There are two x-intercepts: (4, 0) and (-1, 0).
y-intercept:
What value are we going to use for x? You are correct if you said 0.
*Plug in 0 for x
*y-intercept of -4/3
The y-intercept is (0, -4/3).
Note that this will be open hole, as found in step 1.
Step 5: Find and plot several other points on the graph.
5
Note how the vertical asymptote sections the graph into two parts. I’m going to
plug in two x values that are to the left of x = -3.
Plugging in -5 for x we get:
(-5, -18)
Plugging in -4 for x we get:
(-4, -24)
Step 6: Draw curves through the points, approaching the asymptotes.
6
Problem 2. Graph
Answer:
7
x
(x, y)
-2
(-2, .03)
-1
(-1, .17)
0
(0, 1)
1
(1, 6)
2
(2, 36)
Problem 3. Graph
Answer:
8
x
(x, y)
-2
(-2, -36)
-1
(-1, -6)
0
(0, -1)
1
(1, -.17)
2
(2, -.03)
Problem 4.: Find the 1) compound amount AND 2) the compound interest for the
given investment and rate.
1) $25000 for 21 years at an annual rate of 3.25% compounded quarterly.
Answer:
P = 25000
r = 3.25% = .0325
t = 21
n = quarterly = 4 times a year
1)Compound Amount: $49334.23
2) Compound Interest: $24334.23
9
Problems 5a - 5b: Express the given exponential equation in a logarithmic form.
5a.
Answer:
5b.
Answer:
Problems 6a - 6b: Evaluate the given log function without using a calculator.
6a.
Answer:
6b.
10
Answer:
Problems 7a - 7b: Graph the following functions.
7a.
Answer:
11
y
(x, y)
-2
(.01, -2)
-1
(.11, -1)
0
(1, 0)
1
(9, 1)
2
(81, 2)
7b.
Answer:
12
y
(x, y)
-2
(2.001, -2)
-1
(2.012, -1)
0
(2.11, 0)
1
(3, 1)
2
(11, 2)
Problems 8a - 8b: Evaluate the given expression without the use of a calculator.
8a.
Answer:
8b.
Answer:
Problems 9a - 9b: Expand each logarithmic expression as much as possible.
Evaluate without a calculator where possible.
9a.
Answer:
13
9b.
Answer:
Problems 10a - 10b: Condense each logarithmic expression into one logarithmic
expression. Evaluate without a calculator where possible.
10a.
Answer:
10b.
Answer:
14
Problems 11a - 11c: Solve the given exponential equation. Round your answer to
two decimal places.
11a.
Answer:
11b.
Answer:
15
11c.
Answer:
Since e raised to a power cannot equal a negative number, there is only one
solution, x is approx 1.61.
16