Apex Shape

algebra quadratic-functions

Definition

Apex Shape

Apex shape is the representation of a quadratic function in the form

$$f(x)=a(x-d{)}^2+e,$$

where $a

$$S=(d,e).$$

The parameter $a$ determines the vertical scaling and orientation: $a>0$ opens the parabola upwards, while $a<0$ opens it downwards.

Example

Completion of the Square (Transformation into apex shape): Consider the quadratic function

$$f(x)=2x^2+12x+1.$$

Factor out the coefficient of $x^2$ from the quadratic and linear terms:

$$f(x)=2(x^2+6x)+1.$$

Complete the square inside the parentheses:

$$\begin{array}{rl}f(x)& =2(x^2+6x+9-9)+1\\ & =2((x+3{)}^2-9)+1\\ & =2(x+3{)}^2-18+1\\ & =2(x+3{)}^2-17.\end{array}$$

Hence the apex is

$$S=(-3,-17).$$