Precalculus Examples

$f (x) = \log_{2} (x - 9) + 18$

Step 1

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

$g (x) = \log_{2} (x)$

Step 2

The transformation being described is from $g (x) = \log_{2} (x)$ to $f (x) = \log_{2} (x - 9) + 18$ .

$g (x) = \log_{2} (x) \to f (x) = \log_{2} (x - 9) + 18$

Step 3

The transformation from the first equation to the second one can be found by finding $a$ , $c$ and $d$ for $f (x) = \log_{2} (x - 9) + 18$ .

$f (x) = a \log_{b} (x + c) + d$

Step 4

Find $a$ , $c$ and $d$ for $g (x) = \log_{2} (x)$ .

$a_{1} = 1$

$c_{1} = 0$

$d_{1} = 0$

Step 5

Find $a$ , $c$ and $d$ for $f (x) = \log_{2} (x - 9) + 18$ .

$a_{2} = 1$

$c_{2} = - 9$

$d_{2} = 18$

Step 6

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

$f (x) = a \log_{b} (x + c) + d$ - The graph is shifted to the left $c$ units.

$f (x) = a \log_{b} (x - c) + d$ - The graph is shifted to the right $c$ units.

Horizontal Shift: Right $9$ Units

Step 7

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

$f (x) = a \log_{b} (x + c) + d$ - The graph is shifted up $d$ units.

$f (x) = a \log_{b} (x + c) - d$ - The graph is shifted down $d$ units.

Vertical Shift: Up $18$ Units

Step 8

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

Reflection about the x-axis: None

Step 9

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

Reflection about the y-axis: None

Step 10

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

$| a_{2} | > | a_{1} |$ is a vertical stretch (makes it narrower)

$| a_{2} | < | a_{1} |$ is a vertical compression (makes it wider)

Vertical Compression or Stretch: None

Step 11

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, reflection about the y-axis, and if there is a vertical stretch or compression.

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

Horizontal Shift: Right $9$ Units

Vertical Shift: Up $18$ Units

Reflection about the x-axis: None

Reflection about the y-axis: None

Vertical Compression or Stretch: None

Step 12