Question

Identifying Transformations Match the graph with the function. (See the graph $y=\left|x\right|$on page 96.)

$y=\left|x\right|-1$

Short answer

The correct match for$y=\left|x\right|-1$ is I.

Step-by-step solution

Step 1. Apply the concept of transformation of function.

The equation of the given graph is,

$f\left(x\right)=\left|x\right|\mathrm{...}\left(1\right)$

The given equation is,

$g\left(x\right)=\left|x\right|-1$

The above equation can be rewritten as,

role="math" localid="1645165855312" $g\left(x\right)=f\left(x\right)-1\mathrm{...}\left(2\right)$

The horizontal shift of a function is given by,

role="math" localid="1645165925119" $y=f\left(x\right)+d\mathrm{...}\left(3\right)$

If d is positive then the graph will shift upward by d unit at all points.

If d is negative then the graph will shift downward by d unit at all points.

Step 2. Interpret the function g.

Compare equation (2) and (3).

$d=-1$

The value of d is negative, so the graph of g shifts downward by 1 unit with respect to function f.

Step 3. Find the correct match.

The graph that best matches is,

Thus, The correct match for$y=\left|x\right|-1$ is I.