Algebra Examples

f (x) = | x |

$f (x) = | x |$ ,

g (x) = | x | - 4

$g (x) = | x | - 4$

Step 1

The transformation from the first equation to the second one can be found by finding

a

$a$ ,

h

$h$ , and

k

$k$ for each equation.

y = a | x - h | + k

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

Step 2

Factor a

1

$1$ out of the absolute value to make the coefficient of

x

$x$ equal to

1

$1$ .

y = | x |

$y = | x |$

Step 3

Factor a

1

$1$ out of the absolute value to make the coefficient of

x

$x$ equal to

1

$1$ .

y = | x | - 4

$y = | x | - 4$

Step 4

Find

a

$a$ ,

h

$h$ , and

k

$k$ for

y = | x | - 4

$y = | x | - 4$ .

a = 1

$a = 1$

h = 0

$h = 0$

k = - 4

$k = - 4$

Step 5

The horizontal shift depends on the value of

h

$h$ . When

h > 0

$h > 0$ , the horizontal shift is described as:

g (x) = f (x + h)

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

h

$h$ units.

g (x) = f (x - h)

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

h

$h$ units.

Horizontal Shift: None

Step 6

The vertical shift depends on the value of

k

$k$ . When

k > 0

$k > 0$ , the vertical shift is described as:

g (x) = f (x) + k

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

k

$k$ units.

g (x) = f (x) - k

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

k

$k$ units.

Vertical Shift: Down

4

$4$ Units

Step 7

The sign of

a

$a$ describes the reflection across the x-axis.

- a

$- a$ means the graph is reflected across the x-axis.

Reflection about the x-axis: None

Step 8

The value of

a

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

a > 1

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

0 < a < 1

$0 < a < 1$ 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 |

$f (x) = | x |$

Horizontal Shift: None

Vertical Shift: Down

4

$4$ Units

Reflection about the x-axis: None

Vertical Compression or Stretch: None

Step 10