## Precalculus Practice Exam

Part 4 Test 3 Time: 2 hours

1. Given the equation, find the following:

• Amplitude,

• Period,

• Horizontal Shift (phase shift),

Answer: 3 to the left.

• Vertical Shift,

• Domain,

Answer: The set of all real numbers, or in interval notation, .

• Range,

Answer: The range is all values of y between 0 and -20, or in interval notation, [-20,0].

• and four points that have a y-coordinate of 0.

Answer: Here are four such values of x that have a y-coordinate of 0:

There are infinitely many other solutions.

2. Given that the point is located on the terminal side of the angle t, find a point on the terminal side of the following angles:

• -t

3. Write as an algebraic expression in x.

• Convert to radian measure.

• Convert 128.3842 radians to degree measure in the format.

• The graph of is bounded by what two functions?

Answer: The graph of f(x) is bounded by the functions , and .

• Why are these two lines considered boundary functions?

Answer: Because the greatest value can be is 1, and the lowest value it can be is -1. Therefore, the graph of f(x) is contained between the functions , and .

• Graph the function and the two boundary functions.

4. A man, standing on the top of a building that is 1,000 feet high, looks down with an angle of depression of 12 degrees, to the base of the building across the street. How far apart are the buildings?

Answer: The buildings are approximately 4704.63 feet apart.

• Restrict the domain of the function to an interval where the function is increasing, and,

Answer: The biggest interval which will work is .

• Determine over that interval.

• Then sketch the graphs of both f and on the same coordinate system.

• What relationship do the graphs of these two functions have to the graph of the function g(x)=x?

Answer: The graph of is a reflection of the graph of f(x) over the line g(x).

5. A total of \$10,000 is invested for 10 years at 12% per year compounded continuously. How much interest will be earned during the 10 years?

Answer: The interest earned will be \$23,201.17.

6. Solve for x in the inequality

Answer: The solution is the set of all x's on the intervals

• When does .

Answer: When the value of x is greater than 0, x>0.

• When does .

Answer: Both functions are equal when x>0.

7. Assume that the population of a town in Texas is given by , where t time is given in years and t=0 corresponds to the year 1900.
• In what year was the population 60,000?

Answer: It was 60,000 in the year 1915.

• Did the population ever reach 1,000,000?

Answer: It was 1,000,000 in the year 1977.

• How long will it take for the population to reach 10,000?

Answer: The population will never decrease as long as the same growth behavior continues.

8. Find all the zeros of .
• algebraically.

Answer: There must be six (real or complex) zeros since this is a sixth degree polynomial. Using the rational zero test, and synthetic division to find the rational zeros, one obtains as zeros:

• graphically-explain how you interpreted the graph to arrive at your answer.

9. Solve the following system of equations for , and z: