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Quadratic Functions

Quadratic Functions

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Quadratic Functions

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  1. Chapter 11 Quadratic Functions

  2. What is a Quadratic Function? • f(x) = ax² + bx + c, where a ≠ 0 (why a ≠ 0 ?) • A symmetric function that reaches either a maximum or minimum value as x increases • The graph is a parabola • A function in which its solutions can sometimes be found with factoring, can be approximated by graphing, and can always be found using the quadratic formula

  3. Real Life Examples • The trajectory of a baseball thrown into the air, a flare shot from a gun, or a diver jumping off of a cliff • The relationship between the length and width of a rectangle while keeping a constant area • Any symmetric relation that decreases to reach a minimum point and then increases

  4. Name that Parabola • Examples:

  5. Parabola • Symmetric • Line of symmetry • For every y = ax² + bx + c, where a ≠ 0, x = -b / 2a • Vertex • Maximum (sad) • Minimum (happy) • Roots • X-intercept(s)

  6. Roots of a Parabolas • Integral roots (integers) • Estimated roots • One distinct root • No real roots (no x-intercept)

  7. Graphing parabolas • Set the quadratic function equal to zero • Factor if possible • Find the axis of symmetry • Find the coordinates of the vertex • If not factorable, find more points on the graph using a function table. Estimate the roots if applicable

  8. The Quadratic Formula • The solutions of a quadratic equation in the form of ax² + bx + c = 0, where a ≠ 0 are given by the formula x = -b + √ b² - 4ac 2a For ANY Quadratic Function!!!!

  9. The Discriminant and Solutions • √b² - 4ac is positive number • 2 roots • √b² - 4ac is 0 • 1 root • √b² - 4ac is negative number • no solution

  10. Exponential Functions • An Exponential Function is a function that can be described by an equation in the form of y = a˟, where a > 0 and a ≠ 1

  11. Exponential Functions • The change starts out gradual and then becomes much more significant • The graph is a curve • The exponent is the independent variable which varies while the base stays constant • The base describes the growth or decay of the pattern

  12. Property of Equality for Exponential Functions • Suppose a is a positive number other than 1. • Then a˟¹ = a˟² if and only if x₁ = x₂

  13. Real Life Exponential Functions Growth Decay radioactive • Compound interest

  14. Growth and Decay