Algebra Examples

$y = \log (2 x) + 3$

Step 1

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

$y = \log (x)$

Step 2

Remove parentheses.

$y = \log (x)$

Step 3

Assume that $y = \log (x)$ is $f (x) = \log (x)$ and $y = \log (2 x) + 3$ is $g (x) = \log (2 x) + 3$ .

$f (x) = \log (x)$

$g (x) = \log (2 x) + 3$

Step 4

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

$f (x) = \log (x)$

Step 5

The transformation being described is from $f (x) = \log (x)$ to $g (x) = \log (2 x) + 3$ .

$f (x) = \log (x) \to g (x) = \log (2 x) + 3$

Step 6

The transformation from the first equation to the second one can be found by finding $a$ , $c$ and $d$ for $g (x) = \log (2 x) + 3$ .

$g (x) = a \log (x + c) + d$

Step 7

Find $a$ , $c$ and $d$ for $f (x) = \log (x)$ .

$a_{1} = 1$

$c_{1} = 0$

$d_{1} = 0$

Step 8

Find $a$ , $c$ and $d$ for $g (x) = \log (2 x) + 3$ .

$a_{2} = 1$

$c_{2} = 0$

$d_{2} = 3$

Step 9

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

$g (x) = a \log (x + c) + d$ - The graph is shifted to the left $c$ units.

$g (x) = a \log (x - c) + d$ - The graph is shifted to the right $c$ units.

Horizontal Shift: None

Step 10

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

$g (x) = a \log (x + c) + d$ - The graph is shifted up $d$ units.

$g (x) = a \log (x + c) - d$ - The graph is shifted down $d$ units.

Vertical Shift: Up $3$ Units

Step 11

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 12

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 13

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 14

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: $f (x) = \log (x)$

Horizontal Shift: None

Vertical Shift: Up $3$ Units

Reflection about the x-axis: None

Reflection about the y-axis: None

Vertical Compression or Stretch: None

Step 15