# Graph of $y=|x-h|+k$ and $y=-|x-h|+k$ : Exercise (6.1) - Solutions

#### Graph of the Function $y = |x − h| + k$

The graph of the absolute value function$y = |x − h| + k$ can be seen as the translation of $h$-units horizontally and $k$-units vertically of the graph $y = |x|$.

#### Graph of the Function $y = -|x − h| + k$

The graph of the absolute value function$y = -|x − h| + k$ can be seen as the translation of $h$-units horizontally and $k$-units vertically of the graph $y = -|x|$.

1.           Compare the graphs of the following functions to the graph of $y=|x|$.

(a)    $y=|x-3|-2$

(b)    $y=|x+1|+3$

(c)    $y=|x-2|+3$

Show/Hide Solution

(a) The graph of $y=|x-3|-2$ is the translation of positive 3 units horizontally and negative 2 units vertically of the graph $y=|x| .$

(b) The graph of $y=|x+1|+3$ is the translation of negative 1 unit horizontally and positive 3 units vertically of the graph $y=|x| .$

(c) The graph of $y=|x-2|+3$ is the translation of positive 2 units horizontally and positive 3 units vertically of the graph $y=|x| .$

2.           Compare the graphs of the following functions to the graph of $y=-|x|$.

(a)    $y=-|x+3|+2$

(b)    $y=-|x-4|+1$

(c)    $y=-|x+4|-1$

Show/Hide Solution

(a) The graph of $y=-|x+3|+2$ is the translation of negative 3 units horizontally and positive 2 units vertically of the graph $y=-|x| .$

(b) The graph of $y=-|x-4|+1$ is the translation of positive 4 units horizontally and positive 1 unit vertically of the graph $y=-|x| .$

(c) The graph of $y=y=-|x+4|-1$ is the translation of negative 4 units horizontally and negative 1 unit vertically of the graph $y=-|x| .$