Question

the parent function $f(x) = \sqrt3{x}$ is transformed to $g(x) = f(x + 2) - 4$. which is the graph of $g$?

a.
graph a: coordinate plane with a curve labeled g

b.
graph b: coordinate plane with a curve labeled g

Explanation:

Step1: Analyze the transformation of the parent function

The parent function is \( f(x)=\sqrt[3]{x} \). The transformed function is \( g(x)=f(x + 2)-4 \). For a function \( y = f(x) \), the transformation \( y=f(x + h)-k \) means a horizontal shift of \( h \) units to the left (if \( h>0 \)) and a vertical shift of \( k \) units down (if \( k > 0 \)). Here, \( h = 2 \) and \( k=4 \), so the graph of \( f(x) \) is shifted 2 units to the left and 4 units down.

Step2: Find the key point of the parent function

The parent function \( f(x)=\sqrt[3]{x} \) has a key point at \( (0,0) \) (since \( \sqrt[3]{0}=0 \)).

Step3: Apply the transformation to the key point

After shifting 2 units left and 4 units down, the key point \( (0,0) \) of \( f(x) \) will be transformed to \( (0 - 2,0-4)=(- 2,-4) \).

Step4: Analyze the graphs

• For graph A: We can see that the key point (the point where the graph has the "corner" or the inflection point) is around \( (-2,-4) \), which matches our transformed key point.
• For graph B: The key point is around \( (2,4) \) or some other point that does not match the transformed key point.

Answer:

A