Precalculus 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: Down

$5$ 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: Down

$5$ Units

Reflection about the x-axis: None

Vertical Compression or Stretch: None

Step 10

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