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Module 8: Quadratic Functions

Learning Outcomes

  • Identify and find intercepts of quadratic functions for graphing parabolas.

  • Find the y-intercept of a quadratic function by evaluating the function at an input of zero.

  • Determine the x-intercepts (roots) using factoring and the quadratic formula.

Finding Intercepts

  • Y-Intercept: Evaluated by substituting 0 into the function:

    • If f(x) = ax^2 + bx + c, then the y-intercept is f(0) = c.

  • X-Intercepts: Found where the output (f(x)) is zero:

    • Solve the equation f(x) = 0 to find the x-intercepts.

    • The number of x-intercepts varies based on the quadratic function’s graph placement.

Terminology

  • Intercepts: Points where the graph crosses the x-axis (x-intercepts) or the y-axis (y-intercept).

  • Zeros and Roots: Terms commonly used interchangeably with x-intercepts to describe solutions to the quadratic equation.

Steps to Find Intercepts

  1. Y-Intercept:

    • Set x = 0 in the quadratic function f(x).

    • Example: for f(x) = 3x^2 + 5x - 2, compute f(0) = -2.

  2. X-Intercepts:

    • Set f(x) = 0 and solve.

    • Use factoring if possible or apply the quadratic formula when necessary.

Example Calculation

  • For f(x) = 3x^2 + 5x - 2:

    • To find y-intercept: f(0) = -2 (y-intercept is (0, -2)).

    • To find x-intercepts: Solve 3x^2 + 5x - 2 = 0.

Solving Quadratics that Cannot be Factored

  • For quadratics that cannot easily be factored, rewrite in standard form and use vertex form or the quadratic formula:

    • Standard form: f(x) = ax^2 + bx + c.

    • Completing the square or applying the quadratic formula:

      • Quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).

Example of Finding Roots

  • Vertex:

    • Using h = -b/2a, calculate x-coordinate of the vertex.

    • Plug x back into original function to find corresponding y-coordinate (k).

    • Identify roots by setting the function equal to zero and solving.

Real-world Application Example

  • Ball Thrown from a Building:

    • Model: H(t) = -16t^2 + 80t + 40

    • Analyze for:

      • Maximum height (h) at vertex,

      • Time when it reaches maximum height,

      • When it hits the ground (set H(t) = 0).

Practice Problem

  • Model rock thrown from a cliff:

    • H(t) = -16t^2 + 96t + 112

    • Find maximum height, time to reach max height, and descent to ocean.