Question

1. given the cubic function $f(x)=x^3$, which of the following best describes the function $g(x)=(x - 5)^3+2$?

a the same as the graph of $f(x)=x^3$ shifted right by 5 units and up by 2 units
b the same as the graph of $f(x)=x^3$ shifted left by 5 units and up by 2 units
c the same as the graph of $f(x)=x^3$ shifted right by 5 units and down by 2 units
d the same as the graph of $f(x)=x^3$ shifted left by 5 units and down by 2 units

Explanation:

Step1: Recall Function Transformation Rules

For a function \( y = f(x) \), the transformation \( y = f(x - h) + k \) means: if \( h>0 \), shift right by \( h \) units; if \( h<0 \), shift left by \( |h| \) units. For \( k \), if \( k>0 \), shift up by \( k \) units; if \( k<0 \), shift down by \( |k| \) units.

Step2: Analyze \( g(x) = (x - 5)^3 + 2 \)

Compare with \( f(x)=x^3 \), here \( h = 5 \) (so shift right by 5 units) and \( k = 2 \) (so shift up by 2 units).

Answer:

A. The same as the graph of \( f(x) = x^3 \) shifted right by 5 units and up by 2 units