Chapter 2 Polynomial and Rational Functions
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1 SECTION.1 Linear and Quadratic Functions Chapter Polynomial and Rational Functions Section.1: Linear and Quadratic Functions Linear Functions Quadratic Functions Linear Functions Definition of a Linear Function: Graph of a Linear Function: MATH 1330 Precalculus 141
2 CHAPTER Polynomial and Rational Functions Example: 14 University of Houston Department of Mathematics
3 SECTION.1 Linear and Quadratic Functions Solution: Example: Solution: MATH 1330 Precalculus 143
4 CHAPTER Polynomial and Rational Functions 144 University of Houston Department of Mathematics
5 SECTION.1 Linear and Quadratic Functions Parallel and Perpendicular Lines: MATH 1330 Precalculus 145
6 CHAPTER Polynomial and Rational Functions Example: Solution: 146 University of Houston Department of Mathematics
7 SECTION.1 Linear and Quadratic Functions Additional Example 1: Solution: MATH 1330 Precalculus 147
8 CHAPTER Polynomial and Rational Functions Additional Example : Solution: Additional Example 3: 148 University of Houston Department of Mathematics
9 SECTION.1 Linear and Quadratic Functions Solution: Additional Example 4: MATH 1330 Precalculus 149
10 CHAPTER Polynomial and Rational Functions Solution: Quadratic Functions Definition of a Quadratic Function: Graph of a Quadratic Function: 150 University of Houston Department of Mathematics
11 SECTION.1 Linear and Quadratic Functions Example: MATH 1330 Precalculus 151
12 CHAPTER Polynomial and Rational Functions Solution: 15 University of Houston Department of Mathematics
13 SECTION.1 Linear and Quadratic Functions MATH 1330 Precalculus 153
14 CHAPTER Polynomial and Rational Functions Using Formulas to Find the Vertex: Example: Solution: 154 University of Houston Department of Mathematics
15 SECTION.1 Linear and Quadratic Functions Intercepts of the Graph of a Quadratic Function: x-intercepts: MATH 1330 Precalculus 155
16 CHAPTER Polynomial and Rational Functions 156 University of Houston Department of Mathematics
17 SECTION.1 Linear and Quadratic Functions y-intercept: Example: Solution: Note: For a review of factoring, please refer to Appendix A.1: Factoring Polynomials. MATH 1330 Precalculus 157
18 CHAPTER Polynomial and Rational Functions Additional Example 1: Solution: 158 University of Houston Department of Mathematics
19 SECTION.1 Linear and Quadratic Functions MATH 1330 Precalculus 159
20 CHAPTER Polynomial and Rational Functions Additional Example : Solution: 160 University of Houston Department of Mathematics
21 SECTION.1 Linear and Quadratic Functions MATH 1330 Precalculus 161
22 CHAPTER Polynomial and Rational Functions 16 University of Houston Department of Mathematics
23 SECTION.1 Linear and Quadratic Functions Additional Example 3: Solution: MATH 1330 Precalculus 163
24 CHAPTER Polynomial and Rational Functions Additional Example 4: Solution: 164 University of Houston Department of Mathematics
25 SECTION.1 Linear and Quadratic Functions Additional Example 5: Solution: MATH 1330 Precalculus 165
26 CHAPTER Polynomial and Rational Functions 166 University of Houston Department of Mathematics
27 Exercise Set.1: Linear and Quadratic Functions Find the slope of the line that passes through the following points. If it is undefined, state Undefined. 1. (, 3) and (6, 7). ( 1, 6) and ( 5,10) 3. ( 8, 7) and ( 1, 7) 4. ( 3, 8) and (3, 4) Find the slope of each of the following lines. (a) Write the equation in slope-intercept form. (b) Write the equation as a linear function. (c) Identify the slope. (d) Identify the y-intercept. (e) Draw the graph. 11. x y x y x 4y x 5y c 6. d 7. e 8. f e y c x 15. 4x 3y x 1 y 1 3 Find the linear function f that satisfies the given conditions. 17. Slope 4 - ; y-intercept 3 7 f 18. Slope 4 ; y-intercept 5 Find the linear function f which corresponds to each graph shown below. 9. y d 19. Slope 0. Slope ; passes through (-3, ) 9 1 ; passes through (-4, -) 5 1. Passes through (, 11) and (-3, 1) x. Passes through (-4, 5) and (1, -) 3. x-intercept 7; y-intercept x-intercept -; y-intercept 6 5. Slope 3 ; x-intercept y 6. Slope 1 ; x-intercept Passes through (-3, 5); parallel to the line y 1 x 8. Passes through (, -6); parallel to the line y 4 For each of the following equations, 9. Passes through (5, -7); parallel to the line y 5x 3 MATH 1330 Precalculus 167
28 Exercise Set.1: Linear and Quadratic Functions 30. Passes through (5, -7); perpendicular to the line y 5x Passes through (, 3); parallel to the line 5x y 6 3. Passes through (-1, 5); parallel to the line 4x 3y Passes through (, 3); perpendicular to the line 5x y Passes through (-1, 5); perpendicular to the line 4x 3y Passes through (4, -6); parallel to the line containing (3, -5) and (, 1) 36. Passes through (8, 3) ; parallel to the line containing (, 3) and ( 4, 6) 37. Perpendicular to the line containing (4, -) and (10, 4); passes through the midpoint of the line segment connecting these points. 38. Perpendicular to the line containing ( 3, 5) and (7, 1) ; passes through the midpoint of the line segment connecting these points. 39. f passes through 3, 6 and through 8, f passes through, 1 and through 9, 4. 1 f 1 f passes passes 41. The x-intercept for f is 3 and the x-intercept for 1 f is The y-intercept for f is 4 and the y-intercept for 1 f is 6. Answer the following, assuming that each situation can be modeled by a linear function. 43. If a company can make 1 computers for $3,000, and can make 40 computers for $38,00, write an equation that represents the cost of x computers. 44. A certain electrician charges a $40 traveling fee, and then charges $55 per hour of labor. Write an equation that represents the cost of a job that takes x hours. For each of the quadratic functions given below: (a) Complete the square to write the equation in the standard form f ( x) a( x h) k. (b) State the coordinates of the vertex of the parabola. (c) Sketch the graph of the parabola. (d) State the maximum or minimum value of the function, and state whether it is a maximum or a minimum. (e) Find the axis of symmetry. (Be sure to write your answer as an equation of a line.) 45. f ( x) x 6x f ( x) x 8x f ( x) x x 48. f ( x) x 10x 49. f ( x) x 8x f ( x) 3x 18x f ( x) x 8x 9 5. f ( x) x 4x f ( x) 4x 4x f ( x) x 8x f ( x) x 5x f ( x) x 7x f ( x) 3x 4x f ( x) 7 x 3x 168 University of Houston Department of Mathematics
29 Exercise Set.1: Linear and Quadratic Functions Each of the quadratic functions below is written in the form f ( x) ax bx c. For each function: (a) Find the vertex ( hk, ) of the parabola by using b b the formulas k f a a. (Note: When only the vertex is needed, this method can be used instead of completing the square.) (b) State the maximum or minimum value of the function, and state whether it is a maximum or a minimum. 59. f ( x) x 1x f ( x) x 14x f ( x) x 16x 9 h and f ( x) x 16x 40 f ( x) 4x 8x 5 f ( x) 4x 16x 9 f ( x) x 6x 3 f ( x) x 10x 5 f ( x) x x 5 f ( x) x 4 f ( x) 9 4x f ( x) 9x f ( x) 3x 1x f ( x) x 9x f ( x) 6x x 5 The following method can be used to sketch a reasonably accurate graph of a parabola without plotting points. Each of the quadratic functions below is written in the form f ( x) ax bx c. The graph of a quadratic function is a parabola with vertex, where h and k f b a b. a (a) Find all x-intercept(s) of the parabola by setting f( x) 0 and solving for x. (b) Find the y-intercept of the parabola. (c) Give the coordinates of the vertex (h, k) of the parabola, using the formulas h b a and k f b. a (d) State the maximum or minimum value of the function, and state whether it is a maximum or a minimum. (e) Find the axis of symmetry. (Be sure to write your answer as an equation of a line.) (f) Draw a graph of the parabola that includes the features from parts (a) through (d) f ( x) x x 15 f ( x) x 8x 16 For each of the following problems, find a quadratic function satisfying the given conditions. 77. Vertex (, 5) ; passes through ( 7, 70) 78. Vertex ( 1, 8) ; passes through (,10) 79. Vertex ( 5, 7) ; passes through ( 3, 4) 80. Vertex ( 4, 3) ; passes through ( 1,13) Answer the following. 81. Two numbers have a sum of 10. Find the largest possible value of their product. 8. Jim is beginning to create a garden in his back yard. He has 60 feet of fence to enclose the rectangular garden, and he wants to maximize the area of the garden. Find the dimensions Jim should use for the length and width of the garden. Then state the area of the garden. 83. A rocket is fired directly upwards with a velocity of 80 ft/sec. The equation for its height, H, as a function of time, t, is given by the function H ( t) 16t 80t. (a) Find the time at which the rocket reaches its maximum height. (b) Find the maximum height of the rocket. 67. f ( x) 3x 1x 36 MATH 1330 Precalculus 169
30 Exercise Set.1: Linear and Quadratic Functions 84. A manufacturer has determined that their daily profit in dollars from selling x machines is given by the function P( x) 00 50x 0.1x. Using this model, what is the maximum daily profit that the manufacturer can expect? 170 University of Houston Department of Mathematics
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