Chapter 1.6, Problem 56E

Solution

Show that the vertical stretches and shrinks does not change the x -intercept and explain the reason.

The transformation obtained by multiplication of a constant to the function so x -intercept remains unchanged.

Given:

Take an equation $y=f\left(x\right)$

for example whose graph is as follows:

1

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 $c<1$ .

The graph of $y=-f\left(x\right)$ is the reflection of graph of $y=f\left(x\right)$ across x -axis.

Calculation:

Transform the graph of $y=f\left(x\right)$ by a vertical stretch by factor $2$ .

  

Now, transform $y=f\left(x\right)$ by a vertical shrink.by factor $1/2$ .

  

So, in both the case of vertical stretch and shrink, the x -intercept of graph remains unchanged.

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

In the transformation of vertical stretch or shrink the equation is transformed by multiplication of a constant c to the function $f\left(x\right)$ and the x -intercept of a graph is obtained by substituting $y=0$ which is not change due to transformation by multiplication of constant c to the function $f\left(x\right)$ . Hence, the vertical stretches and shrinks does not change the x -intercept.

Conclusion The transformation obtained by multiplication of a constant to the function so x -intercept remains unchanged.