# How do you graph y=log_2(x-5)-3?

Feb 9, 2018

red: $y = {\log}_{2}$
blue: $y = {\log}_{2} \left(x - 5\right) - 3$

#### Explanation:

The point $\left(x , y\right)$ on the red graph corresponds to the point $\left(\frac{x}{d} + h , c y + k\right)$ on the blue graph.

$c = 1$
$d = 1$
$h = 5$
$k = - 3$

Use the points (1,0), (2,1), (4,2), (8,3) on the original graph (red) to transform it to become the blue graph.

$\left(1 , 0\right) \rightarrow \left(\frac{1}{1} + 5 , 1 \left(0\right) - 3\right) = \left(6 , - 3\right)$

$\left(2 , 1\right) \rightarrow \left(\frac{2}{1} + 5 , 1 \left(1\right) - 3\right) = \left(7 , - 2\right)$

$\left(4 , 2\right) \rightarrow \left(\frac{4}{1} + 5 , 1 \left(2\right) - 3\right) = \left(9 , - 1\right)$

$\left(8 , 3\right) \rightarrow \left(\frac{8}{1} + 5 , 1 \left(3\right) - 3\right) = \left(13 , 0\right)$

Plot the new points accordingly and you will get the blue graph.