171S3.3p Analyzing Graphs of Quadratic Functions. October 04, Vertex of a Parabola. The vertex of the graph of f (x) = ax 2 + bx + c is

Transcription

1 MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and Models 3.3 Analyzing Graphs of Quadratic Functions 3.4 Solving Rational Equations and Radical Equations 3.5 Solving Equations and Inequalities with Absolute Value This graphs either one or two quadratic functions; shows the graph, equation, roots, axis of symmetry, and vertex. Graph on left. 3.3 Analyzing Graphs of Quadratic Functions Find the vertex, the axis of symmetry, and the maximum or minimum value of a quadratic function using the method of completing the square. Graph quadratic functions. Solve applied problems involving maximum and minimum function values. View PowerPoint presentation of Quadratic Function. Mathematica Interactive Figures are available through Tools for Success, Activities and Projects in CourseCompass. You may access these through CourseCompass or from the Important Links webpage. You must Login to MML to use this link. Section 3.3 Graphs of Quadratic Functions Section 3.3 Application: Height of a Baseball This graphs one quadratic functions; shows the graph, roots, equation, axis of symmetry, and vertex. Graph on right. cfcc.edu/mathlab/geogebra/quadratic_graph_roots.html Graphing Quadratic Functions of the Type f (x) = a(x h) 2 + k The graph of a quadratic function is called a parabola. The point (h, k) at which the graph turns is called the vertex. The maximum or minimum value of f(x) occurs at the vertex. Each graph has a line x = h that is called the axis of symmetry. Find the vertex, the axis of symmetry, and the maximum or minimum value of f (x) = x x Solution: Complete the square. Login to SAS Curriculum Pathways. Use user name able7oxygen and enter 1445 in the QL # window in the upper right corner. Vertex: ( 5, 2) Axis of symmetry: x = 5, the x value at vertex. Minimum value of the function: 2, the y value at vertex. Graph f (x) = x x + 23 Find the vertex, the axis of symmetry, and the maximum or minimum value of Solution: Complete the square. Vertex of a Parabola The vertex of the graph of f (x) = ax 2 + bx + c is Graph: Vertex: (4, 0) Axis of symmetry: x = 4 Minimum value of the function: 0 We calculate the x coordinate. We substitute to find the y coordinate. The graph of g is a vertical shrinking of the graph of y = x 2 along with a shift of 4 units to the right. 1

2 For the function f(x) = x x 47: a) Find the vertex. b) Determine whether there is a maximum or minimum value and find that value. c) Find the range. d) On what intervals is the function increasing? decreasing? Solution a) f (x) = x x 47 The x coordinate of the vertex is: Since f (7) = (7) (7) 47 = 2, the vertex is (7, 2). b) Since a is negative (a = 1), the graph opens down, so the second coordinate of the vertex, 2, is the maximum value of the function. c) The range is ( 4, 2]. d) Since the graph opens down, function values increase as we approach the vertex from the left and decrease as we move to the right of the vertex. Thus the function is increasing on the interval (, 7) and decreasing on (7, ). Application A stonemason has enough stones to enclose a rectangular patio with 60 ft of stone wall. If the house forms one side of the rectangle, what is the maximum area that the mason can enclose? What should the dimensions of the patio be in order to yield this area? 1. Familiarize. Make a drawing of the situation, using w to represent the width of the fencing. 2. Translate. Since the area of a rectangle is given by length times width, we have A(w) = (60 2w)w = 2w2 + 60w. 3. Carry out. We need to find the maximum value of A(w) and find the dimensions for which that maximum occurs. The maximum will occur at the vertex of the parabola, the first coordinate is Thus, if w = 15 ft, then the length l = = 30 ft and the area is = 450 ft Check. ( ) = 60 feet of fencing. 5. State. The maximum possible area is 450 ft 2 when the patio is 15 feet wide and 30 feet long. 270/6. Given the quadratic function g(x) = x 2 5x + 6. (a) find the vertex; 270/10. Given the quadratic function g(x) = (1/3)x 2 2x + 1. (a) find the vertex; Sep 29 12:11 PM Sep 29 12:11 PM 270/14. Given the quadratic function f(x) = x 2 8x + 5. (a) find the vertex; 270/16. Given the quadratic function f(x) = 3x 2 3x + 1. (a) find the vertex; Sep 29 12:24 PM Sep 29 12:24 PM 2

3 270/18. Match the equation y = (x 4) with one of the graphs (a) (h). 270/22. Match the equation y = (x 3) 2 with one of the graphs (a) (h). 270/24. Match the equation y = 2(x 1) 2 4 with one of the graphs (a) (h). 270/20. Match the equation y = x 2 3 with one of the graphs (a) (h). Sep 29 12:25 PM Sep 29 12:25 PM 271/26. Determine whether the statement is true or false: The vertex of the graph of f(x) = ax 2 + bx + c is b / 2a. 271/27. Determine whether the statement is true or false: The graph of h(x) = (x + 2) 2 can be obtained by translating the graph of h(x) = x 2 right 2 units. 271/34. f(x) = (1/2)x 2 3x + 5/2 271/28. Determine whether the statement is true or false: The vertex of the graph of the function g(x) = 2(x 4) 2 1 is ( 4, 1). 271/30. Determine whether the statement is true or false: The minimum value of the function f(x) = 3(x 1) is 5. Sep 29 12:32 PM 271/36. f(x) = 2x 2 24x /38. f(x) = 3x x 49 3

4 271/40. g(x) = 2x 2 6x /42. Height of a Projectile. A stone is thrown directly upward from a height of 30 ft with an initial velocity of 60 ft/sec. The height of the stone t seconds after it has been thrown is given by the function s(t) = 16t t Determine the time at which the stone reaches its maximum height and find the maximum height. (a) Vertex: (1.5, 0.5) (b) a = 2 > 0, curve opens upward and has a minimum value of 0.5 (c) Range is y > 0.5 or [0.5, ). (d) Increasing: (0.5, ); decreasing (, 0.5) Sep 29 12:38 PM Sep 29 12:40 PM 271/44. Height of a Rocket. A model rocket is launched with an initial velocity of 150 ft / sec from a height of 40 ft. The function s(t) = 16t t + 40 gives the height of the rocket, in feet, t seconds after it has been launched. Determine the time at which the rocket reaches its maximum height and find the maximum height. 271/46. Maximizing Area. A fourth grade class decides to enclose a rectangular garden, using the side of the school as one side of the rectangle. What is the maximum area that the class can enclose with 32 ft of fence? What should the dimensions of the garden be in order to yield this area? Sep 29 12:40 PM Sep 29 12:43 PM 272/48. Maximizing Area. The sum of the base and the height of a parallelogram is 69 cm. Find the dimensions for which the area is a maximum. 272/50. Maximizing Profit. In business, profit is the difference between revenue and cost; that is, Total profit = Total revenue Total cost or P(x) = R(x) C(x), where x is the number of units sold. Find the maximum profit and the number of units that must be sold in order to yield the maximum profit for R(x) = 5x and C(x) = 0.001x x + 60 Sep 29 12:43 PM Sep 29 12:45 PM 4

5 272/52. Maximizing Profit. In business, profit is the difference between revenue and cost; that is, Total profit = Total revenue Total cost or P(x) = R(x) C(x), where x is the number of units sold. Find the maximum profit and the number of units that must be sold in order to yield the maximum profit for R(x) = 20x 0.1x 2 and C(x) = 4x + 2 P(x) = R(x) - C(x) P(x) = (20x - 0.1x 2 ) - (4x + 2) P(x) = -0.1x x - 2 Vertex: (80, 638) Maximum profit is 638 (units of money) when 80 units are sold. Sep 29 12:48 PM 5