Precalculus Examples

$f (x) = \frac{1}{2} {(x + 2)}^{2} - 9$

Step 1

The parent function is the simplest form of the type of function given.

$g (x) = x^{2}$

Step 2

The transformation being described is from $g (x) = x^{2}$ to $f (x) = \frac{1}{2} \cdot {(x + 2)}^{2} - 9$ .

$g (x) = x^{2} \to f (x) = \frac{1}{2} \cdot {(x + 2)}^{2} - 9$

Step 3

The horizontal shift depends on the value of $h$ . The horizontal shift is described as:

$f (x) = f (x + h)$ - The graph is shifted to the left $h$ units.

$f (x) = f (x - h)$ - The graph is shifted to the right $h$ units.

Horizontal Shift: Left $2$ Units

Step 4

The vertical shift depends on the value of $k$ . The vertical shift is described as:

$f (x) = f (x) + k$ - The graph is shifted up $k$ units.

$f (x) = f (x) - k$ - The graph is shifted down $k$ units.

Vertical Shift: Down $9$ Units

Step 5

The graph is reflected about the x-axis when $f (x) = - f (x)$ .

Reflection about the x-axis: None

Step 6

The graph is reflected about the y-axis when $f (x) = f (- x)$ .

Reflection about the y-axis: None

Step 7

Compressing and stretching depends on the value of $a$ .

When $a$ is greater than $1$ : Vertically stretched

When $a$ is between $0$ and $1$ : Vertically compressed

Vertical Compression or Stretch: Compressed

Step 8

Compare and list the transformations.

Parent Function: $g (x) = x^{2}$

Horizontal Shift: Left $2$ Units

Vertical Shift: Down $9$ Units

Reflection about the x-axis: None

Reflection about the y-axis: None

Vertical Compression or Stretch: Compressed

Step 9