Precalculus Practice Exam

Part 4 Test 4 Time: 2 hours

1. Given the equation, find the following:

• Amplitude

Answer: 5.

• Period

Answer:

• Horizontal Shift(phase shift)

Answer: 2 to the right.

• Vertical Shift

Answer: 10 down.

• Domain

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

• Range

Answer: The range is the set of all values of , or in interval notation, [-15,-5].

• and four points that have a y-coordinate of -15.

Answer: Here are such four points that have a y-coordinate of -15:

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:

• Answer: .

• Answer: .

• Answer: .

• -t

Answer: .

3. Write as an algebraic expression in x.

Answer:

• Convert to radian measure.

Answer: 2.6269 radians.

• Convert 150.3040 radians to degree measure in the format.

Answer:

• The graph of is bounded by what two lines?

Answer: The graph of f(x) is bounded by the lines y=2x+1, and y=2x-1.

• Why are these two lines considered boundary lines?

Answer: Because the biggest value can attain is 1, and the lowest value it can attain is -1, the graph of the function is caught between the two boundary lines.

• Graph the function and the two boundary lines.

Answer:

4. A man, standing on the top of a building that is 4,000 feet high, looks down, with an angle of depression of 30 degrees, to the base of the building across the street. How far apart are the buildings?

Answer: The buildings are approximately 6,928 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.

Answer:

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

Answer:

• 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) about the line g(x).

5. A total of \$25,000 is invested for 10 years at 12% per year compounded monthly. How much interest will be earned during the 10 years?

Answer: The interest earned after 10 years will be \$57,509.67.

6. Solve for x in the inequality

Answer: The inequality will be always true except when x=2, and x=-2.

• When does ?

Answer: They are equal only when x>0.

• When does ?

Answer: They are equal only when x>0.

7. Assume that the population of a town in Texas is given by , where t is time 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 1935 with 9 months.

• Did the population ever reach 100? If yes, when? If no, why not?

Answer: It will only reach 100 in the year 2078.

• How long will it take for the population to reach 1,000,000?

Answer: The population can never increase as long as the same decreasing behavior continues.

8. Find the zeros of

• Algebraically

Answer: There are six zeros. Use the rational zero test and synthetic division to find the 4 rational zeros:

The remaining zeros can be found with the quadratic formula.

• Graphically. Explain how you interpreted the graph to arrive at your answer.

Answer: The graph below tells us that there are zeros at , and 4 more roots close to x=3.

Zooming in close to x=3, we detect 2 roots at about x=2.83 and x=3. Since the function is even, there are also roots at about x=-2.83 and x=-3.
9. Solve the following system of equations for and

Answer: .

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Tue Jun 24 23:15:36 MDT 1997

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