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)

identify the graph.

Explanation:

⚡ Using what you learned: combining transformations

Step 1: Identify the key point of the parent function

The parent function is:

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

The inflection point (the "center" of the curve) of the parent function is at:

$$ (0, 0) $$

Step 2: Apply the translations to the key point

The graph is translated left \( 4 \) units and down \( 2 \) units:

• Left \( 4 \) units: \( x \to 0 - 4 = -4 \)
• Down \( 2 \) units: \( y \to 0 - 2 = -2 \)

The new inflection point must be at:

$$ (-4, -2) $$

Step 3: Match with the correct graph

• Graph 1: Shows a cubic shape with an inflection point at \( (-4, -2) \), but the curve is oriented horizontally/reflected incorrectly.
• Graph 2: Shows only a segment of a curve, not the full cubic function.
• Graph 3: Shows a parabola (quadratic function), not a cubic function.
• Graph 4: Shows the correct cubic shape \( y = x^3 \) shifted so that its inflection point is exactly at \( (-4, -2) \).

Answer:

The correct graph is the fourth option (the rightmost graph).