quadratic function

  • Vertex Form Introduction

    • We use the vertex to convert to vertex form and another point (e.g., (0,0)) to calculate the 'a' value.

  • Factored Form

    • General equation:f(x) = a * (x - p) * (x - r)where p and r are x-intercepts.

    • Example intercepts:

      • p = -4

      • r = 0

    • Factored form:f(x) = a * (x + 4) * (x + 0)becomes f(x) = a * x * (x + 4).

    • The calculated 'a' from vertex form is -2.

  • Conversion to Standard Form

    • Start with vertex form:f(x) = -2 * (x + 2)^2.

      • Square:

        • Expand

          • x^2 + 4x + 4

    • Substitute back:

      • Final:f(x) = -2 * (x^2 + 4x + 4)=> f(x) = -2x^2 - 8x - 8

        • Alternative using factored form results in the same standard form:f(x) = -2x^2 - 8x

Verifying Results:

  • y-intercept matches (c-value is 0).

  • Consistent a values confirm correctness.

  • Domain and Range

    • Domain:

      • (-∞, ∞) for all quadratic functions.

    • Range:

      • From the vertex: (-∞, 8] (includes 8).

    • Intervals

      • Increasing: (-∞, -2)

      • Decreasing: (-2, ∞).

  • Vertex Form and X-Intercepts

    • For a graph with one x-intercept, x = 4 leads to vertex form f(x) = a*(x-4)^2.

    • Use a point (e.g. y-intercept) to determine a value when x-intercepts are similar to the vertex.

  • Standard Form from Vertex or Factored

    • For y = a(x-h)^2 + k, expanding results in standard form.

      • For y-intercept at (0, -7), solve using similar steps.

  • Final Example of Quadratic

    • With no x-intercepts, it's vital to use algebra.

    • Points:

      1. (0, -7)

      2. (5, -2)

      3. (10, -7)

    • Factored form based on existing x-intercepts, to calculate vertex position.

  • Conclusion

    • All derived forms should check against calculated values. Ensuring consistency of intercept values and form conversions is critical in final result verification.