Question

f(x) = x³
g(x) = \frac{1}{4}x³ - 2
what transformations have occurred to get the graph of g(x) from the graph of f(x)?
options (purple boxes):
vertical stretch by a factor of 1/4, translation right 2 units
vertical shrink by a factor of 1/4, translation right 2 units
vertical stretch by a factor of 1/4, translation down 2 units
vertical shrink by a factor of 1/4, translation down 2 units

Explanation:

Step1: Identify vertical scaling factor

Compare $f(x)=x^3$ and $g(x)=\frac{1}{4}x^3-2$. The coefficient of $x^3$ is $\frac{1}{4}$, which is a vertical shrink (since $0<\frac{1}{4}<1$) by factor $\frac{1}{4}$.

Step2: Identify vertical translation

The $-2$ at the end of $g(x)$ shifts the graph of the scaled function down 2 units.

Step3: Match to option

The transformations match the last option: vertical shrink by factor $\frac{1}{4}$, translation down 2 units.

Answer:

vertical shrink by a factor of 1/4, translation down 2 units