Algebra and Trigonometry, 5th Edition

3.3 Analyzing Graphs of Quadratic Functions

• Find the vertex, the axis of symmetry, and the maximum or minimum value of a quadratic function using the method of completing the square.

• Graph quadratic functions.

• Solve applied problems involving maximum and minimum function values.

Graphing Quadratic Functions of the Type $f(x)=a{(x-h)}^2+k$

The graph of a quadratic function is called a parabola. The graph of every parabola evolves from the graph of the squaring function $f(x)=x^2$ using transformations.

TRANSFORMATIONS

REVIEW SECTION 2.5

We get the graph of $f(x)=a{(x-h)}^2+k$ from the graph of $f(x)=x^2$ as follows:

$$\begin{array}{rcl}f(x)& =& x^2\\ \downarrow \\ f(x)& =& a{x}^2& \mathbf{\text{Vertical stretching or shrinking with a reflection}}\\ \downarrow & \mathbf{\text{across the }}\mathit{x}\mathbf{\text{-axis if }}\mathit{a}\mathbf{<}\mathbf{0}\\ f(x)& =& a{(x-h)}^2& \mathbf{\text{Horizontal translation}}\\ \downarrow \\ f(x)& =& a{(x-h)}^{}\end{array}$$