Graph f(x) = |x| and g(x) = |x| - 9 in the same coordinate system and describe how the graph of g is related to the graph of f.

Step-by-step solution

1. 1

   Describe the parent graph. $f(x)=|x|$ is the V-shaped absolute-value parent function: vertex at $(0,0)$, slope $-1$ to the left of the vertex and $+1$ to the right. Useful anchor points are $(-2,2)$, $(-1,1)$, $(0,0)$, $(1,1)$, $(2,2)$.

2. 2

   Read the transformation from the algebra. In $g(x)=|x|-9$ the $-9$ is applied outside the absolute value, so it changes the output rather than the input. Changes to the output move the graph vertically.

3. 3

   Apply the shift to each anchor point. Subtracting $9$ from every $y$-value sends $(0,0)\to(0,-9)$, $(1,1)\to(1,-8)$, $(-1,1)\to(-1,-8)$, $(2,2)\to(2,-7)$ and $(-2,2)\to(-2,-7)$.

4. 4

   Note what does not change. No horizontal shift, no reflection, no stretch: the two branches keep slopes $\pm 1$ and the opening still points upward. Only the vertical position moves.

5. 5

   State the relationship. The graph of $g$ is the graph of $f$ translated $9$ units downward, with the vertex relocated from $(0,0)$ to $(0,-9)$.