Math ↠ Quadratic functions ↠ Vertex form

Vertex form

Considering the shifting and stretching, the following general equation, also called the vertex form, arises:

$f(x)=a(x-\color{blue}{d})^2+\color{green}{c}$

From this equation one can see the vertex and how the standard parabola is shifted / stretched. The vertex is always: $S(\color{blue}{d}|\color{green}{c})$

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Remember

The vertex is always the highest or lowest point of a parabola:

  • When the parabola is opened upwards, the vertex is the minimum point.
  • When the parabola is opened downwards, the vertex is the maximum point.

Frequently, however, a quadratic equation exists in the general form $f(x)=ax^2+bx+c$. To convert these to the vertex shape, use completing the square.

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Method

  1. Exclude coefficients before $x^2$
  2. Apply completing the square: $f(x)=x^2+px+(\frac{p}{2})^2-(\frac{p}{2})^2$
  3. Apply binomial formula

Example

Change the function $f(x)=2x^2+8x+6$ to the vertex shape.

  1. Exclude coefficients before $x^2$


    $f(x)=2(x^2+4x+3)$

  2. Apply completing the square


    $f(x)=2(x^2+4x\color{red}{+(\frac{4}{2})^2-(\frac{4}{2})^2}+3)$
    $f(x)=2(x^2+4x+4-4+3)$

  3. Apply binomial formula backwards


    $f(x)=2((x+2)^2-1)$
    $f(x)=2(x+2)^2-2$

  4. Specify vertex

    Note that the sign changes at $d$.

    $S(-2|-2)$