MATHEMATICS 155 COLLEGE ALGEBRA

WORKBOOK

Southeastern Louisiana University Department of Mathematics

Revised Summer 2009

MATHEMATICS 155 WORKBOOK

TABLE OF CONTENTS

Section Page

1.1 Linear Equations 1

1.5 Solving Inequalities 2

1.2 Quadratic Equations 3

1.4 Radical Equations; Equations Quadratic in Form;

Factorable Equations 4

2.1 Distance and Midpoint Formulas 5

2.2 Graphs of Equations in Two Variables;

Intercepts; Symmetry 6

2.3 Lines 7

3.1 Functions 8

3.2 The Graph of a Function 9

3.3 Properties of Functions 10

3.4 Library of Functions; Piecewise-defined Functions 11

3.5 Graphing Techniques; Transformations 12

4.1 Linear Functions and Their Properties 13

4.3 Quadratic Functions and Their Properties 14

5.1 Polynomial Functions and Models 15

5.2 Properties of Rational Functions 16

5.3 The Graph of a Rational Function 17

5.4 Polynomial and Rational Inequalities 18

6.1 Composite Functions 19

6.2 One-to-One Functions; Inverse Functions 20

6.3 Exponential Functions 21

6.4 Logarithmic Functions 22

6.5 Properties of Logarithms 23

6.6 Logarithmic and Exponential Equations 24

6.7 Compound Interest 25

6.8 Exponential Growth and Decay Models 26

8.1 Systems of Linear Equations 27

8.6 Systems of Nonlinear Equations 28

Mathematics 155, Section 1.1 – Linear Equations

Name:

Solve each of the following equations for the given variable.

1. 2(1− v) = 6 2. 4d1 + 6 = 7d1 − 4

3. 14x + 1 = x− 5 4. 3(y − 3

2) + 7y = 1

2y

5.p + 3

4=

2p− 3

56. 2− (x− 3(x− 4)) = 12x

7. Solve for x :ax− b

c + d= 2x 8.

x− 1

x− 18+

12

x2 − 20x + 36=

x + 1

x− 2

1

Mathematics 155, Section 1.5 – Linear Inequalities

Name:

(1) Solve each inequality, and then graph the solution.

(a) 2a− 3 < 15 (b) 4− z ≤ 2

(2) Find the solution to each inequality. Write your answer using interval nota-tion.

(a) 2(x− 1) > 0 (b) 12m− 3 < −7

(c) 25z ≤ 4 (d) 1− r > 9

(e) (2− 3x)−1 < 0 (f) − 3 < 5− (2k + 1) < 10

(3) What is the domain of the variable in the expression√

4x + 3 ?

2

Mathematics 155, Section 1.2 – Quadratic Equations

Name:

Find the real solutions, if any, to each equation.

1. (c + 6)(2c− 1) = 0 2. n2 − 4 = 0

3. (3n− 4)2 = 9 4. t2 − 5t = 0

5. 2q(q − 5) = 8 6. x2 + 4x + 7 = 0

7.5

x + 4= 4 +

3

x− 28. 19− 18d− d2 = 0

3

Mathematics 155, Section 1.4 – Miscellaneous Equations

Name:Find the real solutions of each equation.

(1)√

12− x = x

(2) (a2 − 6a)2 − 2(a2 − 6a)− 35 = 0

(3) s− 2√

s− 3 = 0

(4) 6x3/2 − 5x1/2 = 0

(5) 3x3 + 4x2 = 27x + 36

4

Mathematics 155, Section 2.1 – The Distance and Midpoint Formulas

Name:

(1) Draw an xy-plane and plot the points A(−3, 2) and B(4,−1). Draw the linesegment AB. Find the length of the line segment by constructing a right tri-angle with vertical and horizontal line segments, finding the lengths of thoseline segments, and using the Pythagorean Theorem.

(2) Find all points having a y-coordinate of −3 whose distance from the point(1, 2) is 13.

(3) Find the midpoint of line segment AB from number (1).

(4) The midpoint of the line segment from P1 to P2 is (5,−4). If P2 = (7,−2),what is P1?

5

Mathematics 155, Section 2.2 – Graphs of Equations in Two Variables

Name:

Find the intercepts and graph each equation in (1) and (2) by plotting points. Besure to label the intercepts.

(1) 5x + 2y = 10

(2) 4x2 + y = 4

(3) List the intercepts for the equation y =x2 − 4

2xand test for symmetry.

6

Mathematics 155, Section 2.3 – Lines

Name:

Find an equation of the line for each of the following criteria.

(1) Containing the points (−3, 10) and (2, 12)

(2) Horizontal and containing the point (−50, 62)

(3) Vertical and containing the point (−5, 6)

(4) Passing through the point (4,−1) and perpendicular to the line 2x− 3y = 4

7

Mathematics 155, Section 3.1 – Functions

Name:

(1) Which of the following relations define y as a function of x? EXPLAIN whyor why not for each.

(a) y =√

x (b) x + y2 = 1

(2) Find the domain of the function f(x) =x√

x− 4.

For the functions f(x) = 2x2 + 3 and g(x) = 3x− 4, find:

(3) (f − g)(x)

(4)

(f

g

)(1)

8

Mathematics 155, Section 3.2 – The Graph of a Function

Name:

(1) Use the given graph of the function f to answer parts (a) - (f).

(a) Find f(0) and f(−2). f(0) = f(−2) =

(b) Is f(3) positive or negative?

(c) What is the domain of f?

(d) What is the range of f?

(e) How often does the line y = 1 intersect the graph?

(f) For what value of x does f(x) = 0?

(2) For the function g(x) =2x

x− 2,

(a) Is the point (12,−2

3) on the graph of g?

(b) If x = 4, what is g(x)? What point(s) does this yield?

(c) If g(x) = 1, what is x? What point(s) does this yield?

(d) What is the domain of g?

(e) List the intercepts for the graph of g.

9

Mathematics 155, Section 3.3 – Properties of Functions

Name:

(1) Use the given graph of the function f to answer parts (a) - (e).

(a) Identify the domain and range of f .

(b) Identify the intervals on which f is increasing, decreasing or constant.

(c) Identify the local minima and local maxima.

(d) Is f even, odd, or neither?

(e) Identify the intercepts, if any.

(2) g(x) = −2x2 − 2x

(a) Find the average rate of change from 0 to 3.

(b) Find the equation of the secant line containing (0, g(0)) and (3, g(3)).

10

Mathematics 155, Section 3.4 – Library of Functions + Piecewise-Defined

Name:

(1)

f(x) =

{x + 3, for x ≤ −1,

x2, for x > −1.

Find: f(−4) = f(0) =

f(1) = f(5) =

Graph f(x).

(2)

h(x) =

1x, for x < 0,

2, for x = 0,√x, for x > 0.

Find: h(−2) = h(0) =

h(1) = h(4) =

Graph h(x).

11

Mathematics 155, Section 3.5 – Graphing Techniques; Transformations

Name:

(1) EXPLAIN how the four given transformations in the second equation affectthe graph of f(x) = |x|.

f(x) = −2|x + 25| − 72

(2) The graph of f(x) is given below on the viewing window indicated.

Sketch a graph of each of the following:

(a) y = f(x− 2) (b) y = f(x + 1)− 3

(c) y = −2f(x) (d) y = f(2x)

(3) Find the function that is finally graphed after the following transformationsare applied in order to the graph of y =

√x.

(1) Reflect about the x-axis(2) Shift right 3 units(3) Shift down 2 units

12

Mathematics 155, Section 4.1 – Linear Functions and Their Properties

Name:

(1) Determine the slope and y-intercept for each linear function and graph.

(a) f(x) = 12x− 3 (b) g(x) = −3x + 12

(2) The monthly cost C, in dollars, for international calls on a certain cellularphone plan is a flat rate of 12 dollars plus 38 cents per minute used.(a) Write a linear function that expresses the monthly cost C in terms of thenumber of minutes used, x.

(b) What is the cost if you talk on the phone for 1 hour and 15 minutes?

(c) Suppose that you budget yourself $60 per month for the phone. What isthe maximum number of complete minutes that you can talk?

13

Mathematics 155, Section 4.3 – Quadratic Functions and Their Properties

Name:

(1) Find the vertex of each quadratic graph.

(a) y = x2 − 6x (b) y = x2 + 14x− 1

(c) y = 2x2 − 6x− 13 (d) y = −3x2 − 24x + 18

(2) For f(x) = −(x + 3)2 + 4, give:

Domain: Vertex:

Axis of Symmetry: Range:

x-intercepts: y-intercept:

(3) Give an equation for a parabola which is concave up, has a vertex of (−3,−2),and has an x-intercept of (−5, 0).

14

Mathematics 155, Section 5.1 – Polynomial Functions and Models

Name:

(1) Write equations of polynomial functions which satisfy each set of criteria spec-ified below.

(a) zeros of 2 and −3

(b) zeros of −12, 0 and 5

(c) zeros of 4 and −2 and a y-intercept of 2

(2) Write equations of polynomial functions which could fit each of the graphsshown. Each graph is shown on a standard viewing window. Show your equa-tion in factored form, and give the power function that the graph resemblesfor large values of |x|.

Factored: f(x) = Factored: g(x) =

Power Function: y = Power Function: y =

15

Mathematics 155, Section 5.2 – Properties of Rational Functions

Name:

For each of the following rational functions,

• find and identify the domain,• find and identify the intercepts,• find and identify the vertical and horizontal asymptotes.

(1) f(x) =−x2 + 1

x2 − 5x + 6

Domain:

x-intercepts:

y-intercept:

Equation(s) of asymptote(s):

(2) y =−1

(x + 2)2+ 9

Domain:

x-intercepts:

y-intercept:

Equation(s) of asymptote(s):

(3) Graph y =−1

(x + 2)2+ 9 using transformations and the info found in (2).

16

Mathematics 155, Section 5.3 – The Graph of a Rational Function

Name:

For each of the following rational functions,

• find and identify the domain,• find and identify the intercepts,• find and identify the vertical and horizontal asymptotes,• construct a sign chart to determine where f(x) is positive and negative,• draw a complete graph of f .

(1) f(x) =x

(x− 5)(x + 6)

Domain: x-intercepts:

y-intercept: Eq of asymptote(s):

(2) y =2(x + 4)(x− 6)

x2 − 9

Domain: x-intercepts:

y-intercept: Eq of asymptote(s):

17

Mathematics 155, Section 5.4 – Polynomial and Rational Inequalities

Name:

(1) Find the solution to each inequality. Write your answer using interval nota-tion.

(a) x2 − 9 ≤ 0

(b) 2x3 − x2 > 3x

(c)(x + 5)2

x2 − 4≥ 0

(2) What is the domain of the function f(x) =

√x + 5

8− 2x− x2?

18

Mathematics 155, Section 6.1 – Composite Functions

Name:

(1) Let f(x) = x2 − 9, and g(x) = x + 5. Find the following compositions.

(a) (f ◦ g)(−10) (b) (f ◦ f)(0)

(c) (f ◦ g)(x) (d) (g ◦ f)(x)

(2) A dress store advertised a series of discounts. A discount of 25% was followedby an additional discount of 50%.

(a) Express each discount separately in functional format, labeling the firstdiscount as f(x) and the second discount as s(x).

(b) Express the series of discounts as a composition of the two functionsdesignated in part (a).

(c) Evaluate the composition of the functions for a value of x = $85.

(d) Is the discount described equivalent to a 75% reduction? Explain why orwhy not.

19

Mathematics 155, Section 6.2 – One-to-One and Inverse Functions

Name:

(1) The drawing which follows gives the graph of f(x). Draw the graph of f−1(x)on the same set of axes.

(2) Find f−1(x) given f(x) = 5x3 − 4.

(3) Find g−1(x) given g(x) =x− 4

2x + 1.

20

Mathematics 155, Section 6.3 – Exponential Functions

Name:

(1) Let f(x) = 3x + 2.

(a) What is f(−2)? What point is on the graph of f?

(b) If f(x) = 83, what is x? What point is on the graph of f?

(2) Graph each of the following. Give coordinates of at least 3 points on eachgraph.

(a) y = 3x − 2 (b) y = ex−2

(3) The population of a large U.S. city is growing according to the the functionP = 1, 400, 000(1.023)t, where t is the number of years since 2000.

(a) What was the population in 2000?

(b) What will the population be in 2015 according to this formula?

21

Mathematics 155, Section 6.4 – Logarithmic Functions

Name:

(1) (a) Change ex = 4 to an equivalent expression involving a logarithm.

(b) Change log3 x = 2 to an equivalent expression involving an exponent.

(2) Graph each of the following. Give coordinates of at least 3 points on eachgraph.

(a) y = log2(x + 1) (b) y = ln(x)

(3) Determine the domain, range, and intercept(s) for each of the graphs givenabove. Identify any asymptotes by giving the equations.

(a) For y = log2(x + 1). . . Domain: Range:

Asymptote: Intercept(s):

(b) For y = ln(x). . . Domain: Range:

Asymptote: Intercept(s):

22

Mathematics 155, Section 6.5 – Properties of Logarithms

Name:

(1) Write each expression as a single logarithm. Simplify result as much as pos-sible.

(a) log5(250)− log5(10) (b) 2 log5(a) + log5(2a)

(c) 3 log2(p)− 12log2(p) (d) ln(m2 − 4)− ln(m + 2)

(2) Write each expression as a sum and/or difference of logarithms. Expresspowers as factors. Simplify as much as possible.

(a) log2

(4x

)(b) log3 (27m2)

(c) log

((x− 1)2

x3

)(d) ln (xex)

23

Mathematics 155, Section 6.6 – Logarithmic and Exponential Equations

Name:Solve each equation. Express solutions in exact form.

(1) 3x−2 = 64

(2) log(2x)− log(x− 3) = 1

(3) 2 · 49x − 9 · 7x − 5 = 0

(4) log6(x + 4) + log6(x + 3) = 1

(5) 2x+1 = 51−2x

24

Mathematics 155, Section 6.7 – Compound Interest

Name:

(1) A student wishes to invest $1500 in a savings account yielding 3.5% annualinterest compounded monthly. How much will his investment be worth at theend of 2 years?

(2) You have a credit card which currently holds a balance of $2500. You are notrequired to pay anything for two years (something for a ”special customer”). Ifthe credit card company figures your interest by compounding daily at a rateof 21.9% and if you choose not to make any payments for that time period,what will be your new balance be at the end of the second year?

(3) How many years will it take for an initial investment of $2500 to grow to$5500? Assume a rate of interest of 5.2% compounded continuously?

25

Mathematics 155, Section 6.8 – Exponential Growth and Decay Models

Name:

(1) A radioactive substance decays exponentially at an annual rate given byr = −0.000512. How many grams are left after 200 years from a 10-gramspecimen? Round to the nearest tenth of a gram.

(2) A different radioactive element decays continuously at a rate of 5% per year.If we begin with 20 grams of this element, how long (to the nearest tenth ofa year) will it take for only 10 grams to remain?

(3) At 45◦C, dinitrogen pentoxide (N2O5) decomposes into nitrous dioxide (NO2)and oxygen (O2) according to the law of uninhibited decay. An initial amountof 0.25 M of dinitrogen pentoxide decomposes to 0.15 M in 17 minutes. Howmuch dinitrogen pentoxide will remain after 30 minutes?

26

Mathematics 155, Section 8.1 – Systems of Linear Equations

Name:

(1) Solve the system.x− 2y = 6x + 2y = 30

(2) Solve the system.y = 5− 3x4x− y = 9

(3) My friend and I went out to lunch last week, but we did not pay attention tothe the cost of each item we ordered until we compared receipts. I had onesoft drink and one taco. My bill showed a tax of 15 cents and a total of $2.25.My friend had two soft drinks and three tacos. His bill showed a tax of 36cents and a total of $5.51. How much was each item (before tax)?

27

Mathematics 155, Section 8.6 – Systems of Nonlinear Equations

Name:

Solve each of the following systems.

(1) x + y = 5x2 + y = 5

(2) x− y = 2x2 + y2 = 4

(3) x2 + y2 = 12x2 + y = 10

(4) logx(2y) = 3logx(4y) = 2

28