Math 115 - Mid term

Metropolitan State University

Math 115, College Algebra

Midterm Exam, Summer 2016

Name__________________________________

Submit the completed test to the D2L Midterm Exam folder in the Dropbox no later than

11 PM Saturday, June 25, 2016.

You can use your book, e-book, notes, and calculator. You must not get help from

anyone else, real or virtual. All the work on the test should be your own. Be sure to show

how you solved the problems—there is a penalty for giving answers without justification.

Good luck!

I attest that I completed this exam on my own.

Signature: ________________________________________

Problem 1. (5 points) Evaluate the following arithmetical expression. Show every

single step necessary to find the solution.

6 − 4 ∗ (2 + 32 + 7 ∗ 5 − 16 ÷ 4) − 27 ÷ 3

2

Problem 2. (6 points) Classify each of the following numbers as natural, integer,

rational, irrational, real, or complex. Place each number in as many categories as

possible.

(a.) 4.123456789101112131415

(b.) √−4

(c.) 0

(d.) − 2

5

(e.) −4

(f.) √2

3

Problem 3. (10 points) Calculate the following operations with complex numbers.

Give your final answer in the form 𝑎 + 𝑏𝑖.

(a.) (2 − 3𝑖)(4 + 5𝑖)

(b.) 6−𝑖

2𝑖

(c.) (5𝑖 − 3)(4 + 6𝑖)

(d.) (𝑖 − 1)2

(e.) (𝑥 − 𝑖𝑦)(𝑥 + 𝑖𝑦), where 𝑥 and 𝑦 are real numbers

4

Problem 4. (4 points)

(a.) Expand the algebraic expression (3𝑥 + 2𝑦)2 in the variables 𝑥 and 𝑦. Show every step in the expansion and give your answer in simplest form.

(b.) Find the value of the constant c so that the point (5, 𝑐) lies on the graph of the linear function 𝑓(𝑤) = 6𝑤 − 3. Justify your answer.

Problem 5. (5 points) Find an equation for the line that passes through the points

(0,4) and (3,12).

5

Problem 6. (10 points) Solve the following equations for 𝑥. Show each step necessary to find the solution.

(a.)

3 − [4𝑥 − 7(𝑥 − 2) + 6] = 2𝑥 + 3(𝑥 − 3)

(b.)

𝑎𝑥 − 𝑏 = 𝑏𝑥 − 𝑎

6

Problem 7. (4 points) The following table comes from a function 𝑓

𝑥 𝑓(𝑥)

2 11

4 19

6 25

8 32

Prove that 𝑓 cannot be a linear function. Justify your conclusion.

Problem 8. (6 points) Find an algebraic equation whose graph is the following

line. Justify your answer. Assume the horizontal axis is represented by the

variable 𝑥 and the vertical axis by the variable 𝑦.

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

10

-2 -1 0 1 2 3

7

Problem 9. (10 points) Consider the graph of a function 𝑓 below. Use this graph to complete the work that follows. (Assume the graph does not continue outside of box.)

Estimate…

(a.) The domain of 𝑓

(b.) The range of 𝑓

(c.) The intervals where 𝑓 is increasing

(d.) The relative maxima of 𝑓

(e.) 𝑓(3)

(f.) (𝑓 ∘ 𝑓)(−1)

(g.) We call an input value 𝑐 a fixed point of 𝑓 if 𝑓(𝑐) = 𝑐. Show that 𝑓 has at least three fixed points.

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

8

Problem 10. (10 points) The data in the table below gives the tuition at a state university

for the years 1997-2002.

Year Tuition

1997 9800

1998

1999

10,350

10,800

2000 11,600

2001 12,100

2002 12,700

Show your work and justify your answers in each of the following:

(a.) Find the linear regression function that models the data. Define each variable used.

(b.) Use the function you found to estimate the tuition at the same state university in 2014.

(c.) When do you estimate the tuition will reach $20,000?