Question

shifting the cubic function practice
complete this assessment to review what youve learned. it will not count toward your grade.
let the function be $f(x) = x^3$. which of the following functions represent the transformation of $f(x)$ shifted horizontally to the left 4 units?
option #1: $g(x) = (x - 4)^3$
option #2: $g(x) = (x + 4)^3$
option #3: $g(x) = (x)^3 - 4$
(1 point)

Explanation:

Step1: Recall Horizontal Shift Rule

For a function \( y = f(x) \), a horizontal shift left by \( h \) units (\( h>0 \)) is given by \( y = f(x + h) \).

Step2: Apply Rule to \( f(x)=x^3 \)

We need to shift \( f(x)=x^3 \) left 4 units, so \( h = 4 \). Substituting into the shift formula, we get \( g(x)=f(x + 4)=(x + 4)^3 \).

Step3: Analyze Options

• Option #1: \( g(x)=(x - 4)^3 \) is a shift right 4 units (since it's \( f(x - 4) \)).
• Option #2: \( g(x)=(x + 4)^3 \) matches our derived function for left 4 units shift.
• Option #3: \( g(x)=x^3 - 4 \) is a vertical shift down 4 units, not horizontal.

Answer:

Option #2: \( g(x) = (x + 4)^3 \)