MAT 135 In-Class Assignments Answer Key

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1 MAT 135 In-Class Assignments Answer Key Answers are listed under the heading of each section. Where a section was continued on multiple pages, the answers are all listed under the section heading. If there are mistakes, please let me know and I will replace them. If there are missing pieces, let me know that as well. Jamie Beth Fleischner Introduction to Set Notation 1. List the elements in each set: 2. Find the union of the sets: 3. Find the intersection of the sets: empty set 4. Let A {1,2,3,4,5,6} B {2,4,6,9,11,12}. Are the following statements true or false? T T T T F T T F i) T j) T 5. Write the following in set-builder notation: A = {x x is a natural number and x < 5} A = {x x is an integer and -4 < x < 0} Section 2.1 Solving Linear Equations 1. Solve the following equations: 2. The taxi meter was invented in 1891 by Wilhelm Bruhn. Chicago charges $1.80 plus $0.40 per mile for a taxi ride. 3. In 1992, twice as many people visited their doctor because of a cough than an earache. The total number of doctor s visits for these two ailments was reported to be 45 million.

2 Section 2.2 Formulas and Function 2. For each formula, express y as a function of x, then find y given that x Solve for the indicated variable: Section 2.3 Applications 1) D 2) B 3) C 4) A 5) B 6) D 7) B 8) C 9) C 10) C Section 2.4 Inequalities 1. Solve the inequalities, graph each solution, and write the solution in interval notation: 2. A store selling art supplies finds that x sketch pads are sold each week at a price of p dollars each according to the formula x= p. What price should they charge if they want to sell: 3. What about the solutions to these problems? all real numbers no solution all real numbers no solution

3 Section 2.5 Compound Inequalities 1. Solve the following inequalities. Use a line graph and interval notation to write each solution set Write each union or intersection as a single interval, if possible. empty set Section 2.6 Absolute Value Equations and Inequalities 1) c 2) c 3) 4) b 5) d 6) b 7) c 8) d 9) b 10) c Section 3.1 The Coordinate Plane Find the x- and y-intercepts and graph the line. 3. Complete the ordered pairs so that the equation is satisfied. Section 3.2 Slope of a Line 1. graph 1: ; graph 2: 2. Find the slope of the line through the given points. Then plot each pair of points and draw the line through them parallel: 5.

4 Section 3.3 Equation of a Line 1) a 2) b 3) c 4) a 5) a 6) a 7) c 8) d 9) d 10) a 11) d 12) a 13) b 14) d 15) c 16) Use the table at the right to answer the following questions: no IV, III I, III, IV 17) Write the general form of the equation of a line satisfying the given conditions. 18) 19) Row 1: Row 2: Row 3: Row 4: Row 5: 20) 1 B; 2 c; 3 e; 4 d; 5 a; 6 - f Section 3.5 Functions and Relations 1. Determine whether the relation is a function. Identify the domain and the range. no no 2. Let 2 f (x) 2x 5 and g(x) x 3x 4. Evaluate:

5 3. The function V(t) 3300t 18,000 where V is value and t is time in years can be used to find the value of a large copy machine during the first 5 years of use. $5625 $1500 graph 2.42 years Determine whether the equation represents y as a function of x. yes no Section Solving Systems of Equations 1) a 2) c 3) 288 4) 42; 48 5) Solve the following systems: Parallel lines Same line 6) b 7) c and d 8) b 9) c Section 5.3 Polynomials and Polynomial Functions 1. Evaluate each polynomial for the given value of the variable: Add or subtract as indicated: 0 3) d 4) d 5) b 6) d 7) c 8) 9. Multiply:

6 Section 5.4 Multiplying Binomials 1. Multiply as indicated: 3 i) j) k) 2. For ( and ( below, find f (x) g(x), f (a 2), and g(x 1) g(x) : Section Factoring Polynomials 1. Factor out the greatest common factor: The area (in square meters) of a pool is given by the expression, where l is the length of the pool. 4. Factor: prime i) j) 5. Find the missing factor: 6. Factor, if possible: need to check new original prime

7 7. Factor if possible: i) j) k) l) prime m) Section Factoring Polynomials (continue True or False: 1) T 2) T 3) T 4) T 5) F 6) T 7) T 8) C 9) C 10) c Section 5.8 Solving Equations by Factoring 1. Solve each equation: i) 2. Find all values such that : 3. 7 ½ seconds

8 Section 6.1 Properties of Rational Expressions and Functions 1. Identify the value(s) for which the given expression is undefined: all real numbers Reduce to lowest terms: 4. Fill in the expression that makes the rational expressions equivalent: 5. Fill in the expression that makes the rational expressions equivalent: 6. Find the domain in interval notation: all real numbers Section 6.2 Multiplication and Division of Rational Expressions 1. Perform the indicated operation and express the answers in lowest terms: 2. For 3) b 4) c 5) c 6) c 7)

9 Section 6.3 Addition and Subtraction of Rational Expressions 1) c 2) d 3) d 4) d 5. Perform the indicated operation and simplify where possible: 6. Given 7. The formula Write an expression that represents the perimeter of the figure and simplify. 10. Section 6.5 Division of Polynomials 1. Divide as indicated: 2. Find i) 3. To find the average cost of producing an item, divide the total cost by the number of items produced. A company that manufactures computer disks uses the function to represent the cost of producing x disks. Goes down

10 4. Given same and this is the same as the remainder Section Solving Equations Involving Rational Expressions and Applications 1) c 2) c 3) d 4) b 5) a 6) 7. Solve the following equations: Sec. 7.1 Radicals 1) c 2) b 3) b 4) b 5) a 6) T 7) F 8) T 9) 10) 11. Write each of the expressions in simplified form: 12. Find the value of each root:

11 Sec. 7.2 Rational Exponents 1. Simplify each expression: no i) j) 2. Simplify Sec. 7.3 Adding, Subtracting, and Multiplying Radicals 1. Combine the following expressions, if possible. Assume that any variables under an even root are nonnegative. None i) j) 2. Find True or false: F T T F 5. Multiply or divide as indicated and simplify. Assume that any variables under an even root are nonnegative.

12 6. Find the areas of the figures below: Sec. 7.4 Quotients, Powers, and Rationalizing Denominators 1) b 2) a 3) d 4) d 5) b 6) c 7. Simplify: Sec. 7.5 Solving Equations with Radicals and Exponents 1. b 2. b 3. b 4. c 5. c 6. Solve and check each equation. 7. Solve and check each equation. no solution 8. Sec. 7.6 Complex Numbers 1. Simplify the following as much as possible: 2. Perform the addition or subtraction and write the result in standard form. 3. Perform the operation and write the result in standard form.

13 4. Write the quotient in standard form. 5. Solve: 6. True or false: T F T T T F F 7. Sec. 8.1 Factoring and Completing the Square 1. a 2. d 3. d 4. c 5. b 6. a 7. a 8. b 9. d 10. a 11. c 12. c 13. Given, find all values for which : 14. Solve by completing the square: ; angles are 45

14 Sec The Quadratic Formula 1. Solve using the quadratic formula: 2. Use the discriminant to determine the number and types of solutions for each of the following equations: 2 real numbers 2 real numbers 2 real numbers 2 imaginary numbers 2 real numbers Sec The Quadratic Formula (continue 1) D 2) A 3) B 4) D 5) B 6) A 7) D 8) C 9) A 10) C 11) A 12) Sec. 8.4 Quadratic Functions and Their Graphs 1. Answer the following questions for the graph of shown at the right: 2. Answer the following questions for the graph of shown at the right:

15 3. For the following functions, find the x-intercepts, y-intercept, vertex, domain and range, determine if the graph opens upward or downward, and find any maximum or minimum values. A sketch of the graph is helpful. Vertex Domain Range Up/down Max/min A Down Max B Up Min C Up Min D Up Min 4) C 5) C 6) C 7) C 8) b Sec. 9.1 Graphs of Functions and Relations 1. no III, IV I, III, IV yes 2. Determine whether the relation is a function. Identify the domain and the range. yes no 3. Match the function with the correct graph: I B, II C, III A 3. Given the piecewise function 4. Graph each function and give its domain and range in interval notation: 5. The graph at the right is the graph of vertical line test i) j)

16 Sec. 9.2 Transformation of Graphs 1. Finish the following statements: a. Suppose you know the graph of and you want to graph, where is a positive constant. Then the graph of is exactly the same as the graph of, but gets shrunk. b. Suppose you know the graph of and you want to graph, where is a negative constant. Then the graph of is exactly the same as the graph of, but has shifted down. c. Suppose you know the graph of and you want to graph, where is a negative constant. Then the graph of g(x) is exactly the same as the graph of, but has shifted right. 2. The graph of is shown below. Draw the graph of. Don t forget to label your axes. - graph 3. Name the graphs below. 4. Given the graph of below, draw. Label your axes. - graph 5. Without using the graphing calculator, match the graphs with the functions. - 1C, 2B, 3A, 4E, 5G, 6D, 7H, 8I, 9F Sec Graphs of Polynomial Functions 1. Use the graphs to answer the questions that follow. Note the scales on the axes. vertical line test i) symmetric about the y-axis j) k) 2. Consider the graphs shown below. b. not symmetric around the y-axis or around the origin because c. 3. Match the graph with the polynomial. a-iii, b-ii, c-iv, d-i

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