Quadratic Functions in Vertex Form

Quadratic Functions in Vertex Form

Graphing Quadratic Functions

  • Graphing a quadratic function can be simplified using vertex form.

  • Vertex form of a quadratic function:[ y = a(x - h)^2 + k ]

  • Where:

    • (h, k): The vertex of the parabola.

    • a: Determines the direction of the opening of the parabola (upwards if a > 0, downwards if a < 0).

Converting Standard Form to Vertex Form

  • The given quadratic function is expressed as ( f(x) = -2x^2 - 12x + 6 ).

  • The objective is to transform it into vertex form.

Steps for Transformation

  1. Begin with the quadratic function:[ f(x) = 2x^2 - 4x + 3 ]

  2. Transform into vertex form:[ = -2(x^2 - 4) + 5 ]

Converting Vertex Form to General Form

  • It is also essential to reverse the transformation: turning vertex form back into general form.

  • General form looks like:[ f(x) = ax^2 + bx + c ]

Example Transformations

  • Convert the following into general form:

    1. [ f(x) = -2(x + 3)^2 + 24 ]

    2. [ g(x) = 2(x + 3)^2 - 5 ]

Solving for General Form

  1. From the vertex form: [ f(x) = -2(x + 3)^2 + 24 ]

    • Expand this and rearrange into general form.

Additional Transformations

  • Further transformations can include:

    • Given forms such as [ f(x) = 2(x + 4)^2 - 5 ]

    • Follow the same expansion method to achieve the general format.