Algebra Examples

$f (x) = - 2^{x}$ , $g (x) = 5 - 2^{x}$

Step 1

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

$y = a b^{x - h} + k$

Step 2

Find $a$ , $h$ , and $k$ for $f (x) = - 2^{x}$ .

$a = 1$

$h = 0$

$k = 0$

Step 3

Find $a$ , $h$ , and $k$ for $g (x) = 5 - 2^{x}$ .

$a = - 1$

$h = 0$

$k = 0$

Step 4

The horizontal shift depends on the value of $h$ . 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 5

The vertical shift depends on the value of $k$ . 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: None

Step 6

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: Reflected

Step 7

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 8

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: $f (x) = - 2^{x}$

Horizontal Shift: None

Vertical Shift: None

Reflection about the x-axis: Reflected

Vertical Compression or Stretch: None

Step 9