section3_5 - Section 3.5 Quadratic Functions Page 1 of 14...

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Unformatted text preview: Section 3.5 Quadratic Functions Page 1 of 14 Section 3.5 Quadratic Functions Page 2 of 14 Example 1: For the quadratic function f ( x) x 2 2 x 3, express the function in standard form, sketch the graph, and find the maximum or minimum value. Solution to Example 1: Section 3.5 Quadratic Functions Page 3 of 14 Example 2: For the quadratic function f ( x) 2 x 2 4 x 3, express the function in standard form, sketch the graph, and find the maximum or minimum value. Solution to Example 2: Section 3.5 Quadratic Functions Page 4 of 14 Example 3: Given the quadratic function ( ) 2 3 3, find the vertex and the Section 3.5 Quadratic Functions Page 5 of 14 Section 3.5 Quadratic Functions Page 6 of 14 Example 5: Solution to Example 5: Section 3.5 Quadratic Functions Page 7 of 14 Section 3.5 Quadratic Functions Page 8 of 14 Section 3.5 Quadratic Functions Page 9 of 14 Example 6: Solution to Example 6: Section 3.5 Quadratic Functions Page 10 of 14 Section 3.5 Quadratic Functions Page 11 of 14 Example 7: Solution to Example 7: Section 3.5 Quadratic Functions Page 12 of 14 Example 8: Solution to Example 8: Section 3.5 Quadratic Functions Page 13 of 14 Exercise Set 3.5: Maximum and Minimum Values 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)2 + 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. f ( x) = x 2 + 6 x + 7 f ( x) = x 2 − 8 x + 21 f ( x) = x 2 − 2 x f ( x) = x 2 + 10 x f ( x) = 2 x 2 − 8 x + 11 f ( x) = 3 x + 18 x + 15 f ( x) = − x 2 − 8 x − 9 f ( x) = − x 2 + 4 x − 7 f ( x) = 4 x 2 − 40 x + 115 2 19. f ( x) = −2 x 2 + 16 x − 9 20. f ( x) = 3x 2 − 12 x + 29 21. f ( x) = x 2 + 3x + 1 22. f ( x) = x 2 − 7 x + 2 23. f ( x) = −2 x 2 + 9 x + 3 24. f ( x) = −6 x 2 + x − 5 For each of the following problems, find a quadratic function satisfying the given conditions. 25. Vertex (2, − 5) ; passes through (7, 70) 26. Vertex (−1, − 8) ; passes through (2, 10) 27. Vertex (5, 7) ; passes through (3, 4) 28. Vertex (−4, 3) ; passes through (1, 13) Answer the following. 29. Two numbers have a sum of 10. Find the largest possible value of their product. 30. 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. 31. 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, 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. f ( x) = 5 x 2 − 10 x + 8 11. f ( x) = −2 x 2 − 8 x − 14 12. f ( x) = −4 x 2 + 24 x − 27 13. f ( x) = x 2 − 5 x + 3 14. f ( x) = x 2 + 7 x − 1 15. f ( x) = 2 − 3x − 4 x 2 16. f ( x) = 7 − x − 3x 2 For each of the quadratic functions given below: (a) Find the vertex ( h, k ) of the parabola by using the formulas h = − 2ba and k = f − 2ba . () (b) State the maximum or minimum value of the function, and state whether it is a maximum or a minimum. 17. f ( x) = x 2 − 12 x + 50 18. f ( x) = − x 2 + 14 x − 10 ...
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