Question

how do you translate the graph of (f(x) = x^3) left 4 units and down 2 units? identify the equation of the graph.

• (y = (x - 4)^3 - 2)
• (y = (x + 4)^3 - 2)
• (y = (x + 2)^3 - 4)
• (y = (x - 2)^3 - 4)

Explanation:

⚡ Using what you learned: combining transformations

Step 1: Horizontal Translation

To translate a function \( f(x) \) horizontally to the left by \( h \) units, replace \( x \) with \( (x + h) \):

$$ x \to (x + 4) $$

$$ f(x + 4) = (x + 4)^3 $$

Step 2: Vertical Translation

To translate a function vertically down by \( k \) units, subtract \( k \) from the entire function:

$$ y = f(x + 4) - 2 $$

$$ y = (x + 4)^3 - 2 $$

Answer:

\( y = (x + 4)^3 - 2 \)