MATHEMATICS 155 COLLEGE ALGEBRA WORKBOOK

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

1 Mathematics 155, Section 1.1 Linear Equations Solve each of the following equations for the given variable. 1. 2(1 v) = 6 2. 4d 1 + 6 = 7d 1 4 3. 1 4 x + 1 = x 5 4. 3(y 3 2 ) + 7y = 1 2 y 5. p + 3 4 = 2p 3 5 6. 2 (x 3(x 4)) = 12x 7. Solve for x : ax b c + d = 2x 8. x 1 x 18 + 12 x 2 20x + 36 = x + 1 x 2

2 Mathematics 155, Section 1.5 Linear Inequalities (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 notation. (a) 2(x 1) > 0 (b) 1 2 m 3 < 7 (c) 2 5 z 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?

3 Mathematics 155, Section 1.2 Quadratic Equations Find the real solutions, if any, to each equation. 1. (c + 6)(2c 1) = 0 2. n 2 4 = 0 3. (3n 4) 2 = 9 4. t 2 5t = 0 5. 2q(q 5) = 8 6. x 2 + 4x + 7 = 0 7. 5 x + 4 = 4 + 3 x 2 8. 19 18d d 2 = 0

4 Mathematics 155, Section 1.4 Miscellaneous Equations Find the real solutions of each equation. (1) 12 x = x (2) (a 2 6a) 2 2(a 2 6a) 35 = 0 (3) s 2 s 3 = 0 (4) 6x 3/2 5x 1/2 = 0 (5) 3x 3 + 4x 2 = 27x + 36

5 Mathematics 155, Section 2.1 The Distance and Midpoint Formulas (1) Draw an xy-plane and plot the points A( 3, 2) and B(4, 1). Draw the line segment AB. Find the length of the line segment by constructing a right triangle with vertical and horizontal line segments, finding the lengths of those line 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 P 1 to P 2 is (5, 4). If P 2 = (7, 2), what is P 1?

6 Mathematics 155, Section 2.2 Graphs of Equations in Two Variables Find the intercepts and graph each equation in (1) and (2) by plotting points. Be sure to label the intercepts. (1) 5x + 2y = 10 (2) 4x 2 + y = 4 (3) List the intercepts for the equation y = x2 4 2x and test for symmetry.

7 Mathematics 155, Section 2.3 Lines 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

8 Mathematics 155, Section 3.1 Functions (1) Which of the following relations define y as a function of x? EXPLAIN why or why not for each. (a) y = x (b) x + y 2 = 1 (2) Find the domain of the function f(x) = x x 4. For the functions f(x) = 2x 2 + 3 and g(x) = 3x 4, find: (3) (f g)(x) (4) ( ) f (1) g

9 Mathematics 155, Section 3.2 The Graph of a Function (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 ( 1, 2 ) on the graph of g? 2 3 (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.

10 Mathematics 155, Section 3.3 Properties of Functions (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) = 2x 2 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)).

11 Mathematics 155, Section 3.4 Library of Functions + Piecewise-Defined (1) f(x) = { x + 3, for x 1, x 2, for x > 1. Find: f( 4) = f(0) = f(1) = f(5) = Graph f(x). (2) 1, for x < 0, x h(x) = 2, for x = 0, x, for x > 0. Find: h( 2) = h(0) = h(1) = h(4) = Graph h(x).

12 Mathematics 155, Section 3.5 Graphing Techniques; Transformations (1) EXPLAIN how the four given transformations in the second equation affect the 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 transformations are 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

13 Mathematics 155, Section 4.1 Linear Functions and Their Properties (1) Determine the slope and y-intercept for each linear function and graph. (a) f(x) = 1 x 3 (b) g(x) = 3x + 12 2 (2) The monthly cost C, in dollars, for international calls on a certain cellular phone 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 the number 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 is the maximum number of complete minutes that you can talk?

14 Mathematics 155, Section 4.3 Quadratic Functions and Their Properties (1) Find the vertex of each quadratic graph. (a) y = x 2 6x (b) y = x 2 + 14x 1 (c) y = 2x 2 6x 13 (d) y = 3x 2 24x + 18 (2) For f(x) = (x + 3) 2 + 4, give: Domain: Axis of Symmetry: x-intercepts: Vertex: Range: 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).

15 Mathematics 155, Section 5.1 Polynomial Functions and Models (1) Write equations of polynomial functions which satisfy each set of criteria specified below. (a) zeros of 2 and 3 (b) zeros of 1 2, 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 graphs shown. Each graph is shown on a standard viewing window. Show your equation in factored form, and give the power function that the graph resembles for large values of x. Factored: f(x) = Factored: g(x) = Power Function: y = Power Function: y =

16 Mathematics 155, Section 5.2 Properties of Rational Functions 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 x 2 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 + 9 using transformations and the info found in (2). (x + 2) 2

17 Mathematics 155, Section 5.3 The Graph of a Rational Function 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) = Domain: x (x 5)(x + 6) x-intercepts: y-intercept: Eq of asymptote(s): (2) y = Domain: 2(x + 4)(x 6) x 2 9 y-intercept: x-intercepts: Eq of asymptote(s):

18 Mathematics 155, Section 5.4 Polynomial and Rational Inequalities (1) Find the solution to each inequality. Write your answer using interval notation. (a) x 2 9 0 (b) 2x 3 x 2 > 3x (c) (x + 5) 2 x 2 4 0 (2) What is the domain of the function f(x) = x + 5 8 2x x 2?

19 Mathematics 155, Section 6.1 Composite Functions (1) Let f(x) = x 2 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 followed by an additional discount of 50%. (a) Express each discount separately in functional format, labeling the first discount as f(x) and the second discount as s(x). (b) Express the series of discounts as a composition of the two functions designated 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 or why not.

20 Mathematics 155, Section 6.2 One-to-One and Inverse Functions (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) = 5x 3 4. (3) Find g 1 (x) given g(x) = x 4 2x + 1.

21 Mathematics 155, Section 6.3 Exponential Functions (1) Let f(x) = 3 x + 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 each graph. (a) y = 3 x 2 (b) y = e x 2 (3) The population of a large U.S. city is growing according to the the function P = 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?

22 Mathematics 155, Section 6.4 Logarithmic Functions (1) (a) Change e x = 4 to an equivalent expression involving a logarithm. (b) Change log 3 x = 2 to an equivalent expression involving an exponent. (2) Graph each of the following. Give coordinates of at least 3 points on each graph. (a) y = log 2 (x + 1) (b) y = ln(x) (3) Determine the domain, range, and intercept(s) for each of the graphs given above. Identify any asymptotes by giving the equations. (a) For y = log 2 (x + 1)... Domain: Range: Asymptote: Intercept(s): (b) For y = ln(x)... Domain: Range: Asymptote: Intercept(s):

23 Mathematics 155, Section 6.5 Properties of Logarithms (1) Write each expression as a single logarithm. Simplify result as much as possible. (a) log 5 (250) log 5 (10) (b) 2 log 5 (a) + log 5 (2a) (c) 3 log 2 (p) 1 2 log 2(p) (d) ln(m 2 4) ln(m + 2) (2) Write each expression as a sum and/or difference of logarithms. Express powers as factors. Simplify as much as possible. (a) log 2 ( 4 x ) (b) log 3 (27m 2 ) ( ) (x 1) 2 (c) log x 3 (d) ln (xe x )

24 Mathematics 155, Section 6.6 Logarithmic and Exponential Equations Solve each equation. Express solutions in exact form. (1) 3 x 2 = 64 (2) log(2x) log(x 3) = 1 (3) 2 49 x 9 7 x 5 = 0 (4) log 6 (x + 4) + log 6 (x + 3) = 1 (5) 2 x+1 = 5 1 2x

25 Mathematics 155, Section 6.7 Compound Interest (1) A student wishes to invest $1500 in a savings account yielding 3.5% annual interest compounded monthly. How much will his investment be worth at the end of 2 years? (2) You have a credit card which currently holds a balance of $2500. You are not required to pay anything for two years (something for a special customer ). If the credit card company figures your interest by compounding daily at a rate of 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?

26 Mathematics 155, Section 6.8 Exponential Growth and Decay Models (1) A radioactive substance decays exponentially at an annual rate given by r = 0.000512. How many grams are left after 200 years from a 10-gram specimen? 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 of a year) will it take for only 10 grams to remain? (3) At 45 C, dinitrogen pentoxide (N 2 O 5 ) decomposes into nitrous dioxide (NO 2 ) and oxygen (O 2 ) according to the law of uninhibited decay. An initial amount of 0.25 M of dinitrogen pentoxide decomposes to 0.15 M in 17 minutes. How much dinitrogen pentoxide will remain after 30 minutes?

27 Mathematics 155, Section 8.1 Systems of Linear Equations (1) Solve the system. x 2y = 6 x + 2y = 30 (2) Solve the system. y = 5 3x 4x y = 9 (3) My friend and I went out to lunch last week, but we did not pay attention to the the cost of each item we ordered until we compared receipts. I had one soft 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 36 cents and a total of $5.51. How much was each item (before tax)?

28 Mathematics 155, Section 8.6 Systems of Nonlinear Equations Solve each of the following systems. (1) x + y = 5 x 2 + y = 5 (2) x y = 2 x 2 + y 2 = 4 (3) x 2 + y 2 = 12 x 2 + y = 10 (4) log x (2y) = 3 log x (4y) = 2