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Understanding Quadratic Functions in Mathematics

Learn about quadratic functions, identify vertices, x-intercepts, and graph properties. Practice examples included.

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Understanding Quadratic Functions in Mathematics

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  1. Quadratic Functions (Section 2-1)

  2. Section 2.1,Definition of Polynomial Function

  3. f(x) = a is the constant function • f(x) = mx + b where m≠0 is a linear function • f(x) = ax2 + bx + c with a≠0 is a quadratic function • The graph of a quadratic function is a U-shaped curve called a parabola. • The axis of symmetry of the parabola is the line about which the parabola is symmetric. • The vertex of the parabola is the point where the axis of symmetry intersects the parabola. • If a > 1, the graph opens upwards. If a < 1 , the graph opens downwards. • a > 1 is a vertical stretch and 0 < a < 1 is a vertical shrink.

  4. Example 1 Describe how the graph of each function is related to the graph of y = x2? a) b) c) d)

  5. Standard Form of the Quadratic Equation f(x) = a(x-h)2 + k a≠0, where the Axis of Symmetry: line x = h and the Vertex: (h, k) To find the x-intercepts of the graph solve the equation ax2 + bx + c = 0.

  6. Sketch the graph of the quadratic function. Identify the vertex and x-intercepts. Example 2 f(x) = 64 – x2

  7. Sketch the graph of the quadratic function. Identify the vertex and x-intercepts. Example 3 f(x) = ( x – 3)2 + 6

  8. Sketch the graph of the quadratic function. Identify the vertex and x-intercepts. Example 4 f(x) = x2 – 6x + 8

  9. Sketch the graph of the quadratic function. Identify the vertex and x-intercepts. Example 5 f(x) = – x2 +6x – 8

  10. Sketch the graph of the quadratic function. Identify the vertex and x-intercepts. Example 6 f(x) = 2x2 + 8x + 7

  11. HW # 19 pg 99 (1-13 odd, 15-27 odd)

  12. Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Verify your result with a graphing utility. Example 7 Vertex: (1, 2) Point: (3, -6)

  13. Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Verify your result with a graphing utility. Example 8 Vertex: (-4, 11) Point: (-6, 15)

  14. Determine the x-intercept(s) of the graph visually. How do the x-intercepts correspond to the solutions of the quadratic equation when y = 0? Example 9 y = x2 + x – 2

  15. Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x-intercepts. (There are many correct answers.) Example 10

  16. Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x-intercepts. (There are many correct answers.) Example 11

  17. Minimum and Maximum values of Quadratic Function ax2 + bx + c = 0 If a > 0, f has a minimum value at If a < 0, f has a maximum value at

  18. Example 12 (example 5 in book) A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and at an angle of 45 with respect to the ground. The path of the baseball is given by the function f(x) = -0.0032x2 + x + 3, where f(x) is the height of the baseball (in feet) and x is the horizontal distance from home plate (in feet). What is the maximum height reached by the baseball?

  19. Example 13 A football is thrown at a point 6 feet above the ground at a velocity of 60 feet per second and at an angle of 45 with respect to the ground. The path of the baseball is given by the function f(x) = -0.0168x2 + x + 6, where f(x) is the height of the football (in feet) and x is the horizontal distance from the quarterback(in feet). What is the maximum height reached by the football?

  20. Example 14 (example 6 in book) A soft drink manufacturer has daily production costs of C(x) = 70,000 – 120x + 0.055x2 Where C is the total cost (in dollars) and x is the number of units produced. Estimate numerically the number of units that should be produced each day to yield a minimum cost.

  21. Example 15 A local newspaper has daily production costs of C = 55000 - 108x + 0.06x2 where C is the total cost (in dollars) and x is the number of newspapers printed. How many newspapers should be printed each day to yield a minimum cost?

  22. Find the value of b such that the function has the given maximum or minimum value. Example 16 f(x) = -x2 + bx – 16 Maximum value: 48

  23. HW # 20 pg 100-102 (29 – 47 odd, 55, 57, 59, 65, 67, 69, 73, 75)

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