Chapter 1.6, Problem 49E

Solution

The graph G that is obtained by sequence of transformation in graph of y.

The graph G is $y=2\left|x+2\right|-4$ .

Given:

The graph of $y=\left|x\right|$ is shifted by $2$ units left, then a vertical stretch by a factor $2$ and finally a down shift of $4$ units.

Concept Used:

The graph of $y=f\left(x\right)$ is translated or shifted horizontally left or right according as:

1) If $y=f\left(x-c\right)$ , then graph shifted by c units right from $y=f\left(x\right)$ .

2) If $y=f\left(x+c\right)$ , then graph shifted by c units left from $y=f\left(x\right)$ .

And, the graph of $y=f\left(x\right)$ is translated or shifted vertically up or down according as:

1) If $y=f\left(x\right)+c$ , then graph shifted by c units up from $y=f\left(x\right)$ .

2) If $y=f\left(x\right)-c$ , then graph shifted by c units down from $y=f\left(x\right)$ .

The transformation of horizontal stretches or shrinks from $y=f\left(x\right)$ is as follows:

The graph of $y=f\left(\frac{x}{c}\right)$ transform to horizontal stretches by factor c , when $c>1$ and horizontal shrinks by factor c when $c<1$ .

The transformation of vertical stretches or shrinks from $y=f\left(x\right)$ is as follows:

The graph of $y=c\cdot f\left(x\right)$ transform to vertical stretches by factor c , when $c>1$ and shrinks by factor c when $c<1$ .

Calculation:

Consider the sequence of transformation,

The graph of $y=\left|x\right|$ is shifted by $2$ units left,

So x is replace by $\left(x+2\right)$ , that is, $y=\left|x+2\right|$ .

Thus, the equation $y=\left|x\right|$ transform to $y=\left|x+2\right|$ .

And, in the next transformation, the graph of $y=\left|x+2\right|$ is stretches vertically by factor $2$ ,

So $2$ is multiplied to the function, that is $y=2\left|x+2\right|$ .

Hence, the equation $y=\left|x+4\right|$ transform to $y=2\left|x+2\right|$ .

The next transformation, the graph of $y=2\left|x+2\right|$ shifted down by $4$ units,

So $4$ is subtracted from left side of function, that is $y=2\left|x+2\right|-4$ .

Hence, the equation $y=2\left|x+2\right|$ transform to $y=2\left|x+2\right|-4$ .

Conclusion The equation $y=\left|x\right|$ transform to $y=2\left|x+2\right|-4$ .