Compare the graph of the following function with the graph of $f ( x ) = x ^ { 3 }$.
$$y = [ 5 ( x + 10 ) ] ^ { 3 }$$
A) vertical shift of 10 units up
B) vertical shift of 10 units up horizontal shrink of $\frac { 1 } { 5 }$ units
C) horizontal shift of 10 units to the left horizontal shrink of $\frac { 1 } { 125 }$ units
D) horizontal shift of 10 units to the left horizontal stretch of $\frac { 1 } { 5 }$ units
E) horizontal shift of 10 units to the left vertical shift of 5 units up

Question

Quizplus · Accepted Answer

The function $y = [ 5 ( x + 10 ) ] ^ { 3 }$ represents a horizontal shift of 10 units to the left due to the $x + 10$ term. The multiplication by 5 inside the cube does not affect the horizontal stretch/shrink directly; instead, it affects the steepness of the graph. The cube of 5 ($5^3$) amplifies the output by a factor of 125, but this is not a horizontal shrink or stretch; it affects the vertical scale. Thus, the only correct transformation described is the horizontal shift of 10 units to the left. The options mentioning vertical shifts, horizontal shrinks, or stretches in terms of $\frac{1}{5}$ or $\frac{1}{125}$ units do not accurately describe the transformation from $f(x) = x^3$ to $y = [5(x + 10)]^3$.

Compare the Graph of the Following Function with the Graph f(x)=x3f ( x ) = x ^ { 3 }f(x)=x3

Compare the Graph of the Following Function with the Graph $f ( x ) = x ^ { 3 }$