Examples

$f (x) = | x |$ , $g (x) = | x | + 5$

Step 1

The transformation from the first equation to the second one can be found by finding $a$ , $h$ , and $k$ for each equation.

$y = a | x - h | + k$

Step 2

Factor a $1$ out of the absolute value to make the coefficient of $x$ equal to $1$ .

$y = | x |$

Step 3

Factor a $1$ out of the absolute value to make the coefficient of $x$ equal to $1$ .

$y = | x | + 5$

Step 4

Find $a$ , $h$ , and $k$ for $y = | x | + 5$ .

$a = 1$

$h = 0$

$k = 5$

Step 5

The horizontal shift depends on the value of $h$ . When $h > 0$ , the horizontal shift is described as:

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

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

Horizontal Shift: None

Step 6

The vertical shift depends on the value of $k$ . When $k > 0$ , the vertical shift is described as:

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

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

Vertical Shift: Up $5$ Units

Step 7

The sign of $a$ describes the reflection across the x-axis. $- a$ means the graph is reflected across the x-axis.

Reflection about the x-axis: None

Step 8

The value of $a$ describes the vertical stretch or compression of the graph.

$a > 1$ is a vertical stretch (makes it narrower)

$0 < a < 1$ is a vertical compression (makes it wider)

Vertical Compression or Stretch: None

Step 9

To find the transformation, compare the two functions and check to see if there is a horizontal or vertical shift, reflection about the x-axis, and if there is a vertical stretch.

Parent Function: $f (x) = | x |$

Horizontal Shift: None

Vertical Shift: Up $5$ Units

Reflection about the x-axis: None

Vertical Compression or Stretch: None

Step 10

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