Chapter 1.6, Problem 62E

Solution

The correct choice from the given options that represents a horizontal translation of $4$ units left for a function f .

Hence, (C) is the correct choice from the given multiple choices that represents a horizontal translation of $4$ units left for a function f .

Given:

The multiple choices are given as:

(A) $y=f\left(x\right)+4$

(B) $y=f\left(x\right)-4$

(C) $y=f\left(x+4\right)$

(D) $y=f\left(x-4\right)$

(E) $y=4f\left(x\right)$

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:

Consider the given choices,

(A) $y=f\left(x\right)+4$

Since, the graph of $y=f\left(x\right)+c$ translate vertical shift by c units up from $y=f\left(x\right)$ .

The equation $y=f\left(x\right)+4$ represents a shift of $4$ units up.

Hence, (A) is not the correct choice from the given multiple choices that represents a horizontal translation of $4$ units left for a function f .

Consider choice (B) $y=f\left(x\right)-4$ .

Since, the graph of $y=f\left(x\right)-c$ translate vertical shift by c units down from $y=f\left(x\right)$ .

The equation $y=f\left(x\right)-4$ represents a shift of $4$ units down.

Hence, (B) is not the correct choice from the given multiple choices that represents a horizontal translation of $4$ units left for a function f .

Consider choice (C) $y=f\left(x+4\right)$ .

Since, the graph of $y=f\left(x+c\right)$ translate horizontal shift by c units left from $y=f\left(x\right)$ .

The equation $y=f\left(x+4\right)$ represents a shift of $4$ units left.

Hence, (C) is the correct choice from the given multiple choices that represents a horizontal translation of $4$ units left for a function f .

Consider choice (D) $y=f\left(x-4\right)$ .

Since, the graph of $y=f\left(x-c\right)$ translate horizontal shift by c units right from $y=f\left(x\right)$ .

The equation $y=f\left(x+4\right)$ represents a shift of $4$ units right.

Hence, (D) is not the correct choice from the given multiple choices that represents a horizontal translation of $4$ units left for a function f .

Consider choice (E) $y=4f\left(x\right)$ .

Since, 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 equation $y=4f\left(x\right)$ represents vertical stretch by factor $4$ .

Means, choice (E) is not correct.

Hence, (C) is the correct choice from the given multiple choices that represents a horizontal translation of $4$ units left for a function f .

Conclusion

The correct choice is (C).