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Section 31 Quadratic Functions

Section 31 Quadratic Functions. JMerrill, 05 Revised 08. Definition of a Quadratic Function. Let a, b, and c be real numbers with a ≠ 0. The function given by f(x) = ax 2 + bx + c is called a quadratic function

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Section 31 Quadratic Functions

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  1. Section 31Quadratic Functions JMerrill, 05 Revised 08

  2. Definition of a Quadratic Function • Let a, b, and c be real numbers with a ≠ 0. The function given by f(x) = ax2 + bx + c is called a quadratic function • Your book calls this “another form”, but this is the standard form of a quadratic function.

  3. The graph of a quadratic equation is a Parabola. Parabolas occur in many real-life situations All parabolas are symmetric with respect to a line called the axis of symmetry. The point where the axis intersects the parabola is the vertex. Parabolas vertex

  4. Graph of f(x)=ax2, a > 0 Domain (- ∞, ∞) Range [0, ∞) Decreasing (- ∞, 0) Increasing (0, ∞) Zero/Root/solution (0,0) Orientation Opens up Characteristics

  5. Graph of f(x)=ax2, a > 0 Domain (- ∞, ∞) Range (-∞, 0] Decreasing (0, ∞) Increasing (-∞, 0) Zero/Root/solution (0,0) Orientation Opens down Characteristics

  6. Max/Min • A parabola has a maximum or a minimum min max

  7. Vertex Form • The vertex form of a quadratic function is given by: f(x) = a(x – h)2 + k, a ≠ 0 • In this parabola: • the axis of symmetry is x = h • The vertex is (h, k) • If a > o, the parabola opens upward. If a < 0, the parabola opens downward.

  8. Example • In the equation f(x) = -2(x – 3)2 + 8, the graph: • Opens down • Has a vertex at (3, 8) • Axis of Symmetry: x = 3 • Has zeros at • 0 = -2(x – 3)2 + 8 • -8 = -2(x – 3)2 • 4 = (x – 3)2 • 2 = x – 3 or -2 = x – 3 • X = 5 x = 1

  9. Vertex Form from Standard Form • Describe the graph of f(x) = x2 + 8x + 7 • In order to do this, you have to complete the square to put the problem in vertex form Opens Up (-4, -9) Vertex? Orientation?

  10. You Do • Describe the graph of f(x) = x2 - 6x + 7 Opens Up Vertex? (3, -2) Orientation?

  11. Example • Describe the graph of f(x) =2x2 + 8x + 7 Opens Up Vertex? (-2, -1) Orientation?

  12. You Do • Describe the graph of f(x) =3x2 + 6x + 7 Opens Up Vertex? (-1, 4) Orientation?

  13. Write the vertex form of the equation of the parabola whose vertex is (1,2) and passes through (3, - 6) (h,k) = (1,2) Since the parabola passes through (3, -6), we know that f(3) = - 6. So:

  14. Finding Minimums/Maximums • If a > 0, f has a minimum at • If a < 0, f has a maximum at • Ex: a baseball is hit and the path of the baseball is described by • f(x)= -0.0032x2 + x + 3. What is the maximum height reached by the baseball? Remember the quadratic model is: ax2+bx+c F(x)= - 0.0032(156.25)2+156.25+3 = 81.125 feet

  15. Maximizing Area

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