Question

if $f(x) = x^3$, what is the equation of the graphed function? graph of a cubic function with axes labeled x and y, passing through a point near (-2, -2) and other points as shown
a. $y = f(x - 3) - 2$
b. $y = f(x + 3) - 2$
c. $y = f(x + 2) - 3$
d. $y = f(x - 2) + 3$

Explanation:

Step1: Recall function transformation rules

For a function \( y = f(x) \), the transformation \( y = f(x - h) + k \) represents a horizontal shift by \( h \) units (right if \( h>0 \), left if \( h<0 \)) and a vertical shift by \( k \) units (up if \( k>0 \), down if \( k<0 \)).

Step2: Analyze the graph of \( f(x)=x^3 \)

The parent function \( f(x)=x^3 \) has a point at \( (0,0) \). The graphed function seems to have a point that is shifted. Let's check the shifts. From the graph, we can see the behavior: compared to \( f(x)=x^3 \), the graph is shifted left (since the inflection point or key point is moved left) and down.

Step3: Check each option

• Option A: \( y = f(x - 3)-2 \) is a shift right 3 and down 2. Not matching.
• Option B: \( y = f(x + 3)-2 \) is a shift left 3 (since \( h=-3 \)) and down 2. Let's verify. If \( f(x)=x^3 \), then \( f(x + 3)=(x + 3)^3 \), and \( y=(x + 3)^3-2 \). The shift left 3 and down 2 matches the graph's behavior (the key point of \( f(x) \) at \( (0,0) \) would move to \( (-3,-2) \), which seems consistent with the graph).
• Option C: \( y = f(x + 2)-3 \) is shift left 2 and down 3. The down shift is more than what's seen.
• Option D: \( y = f(x - 2)+3 \) is shift right 2 and up 3. Opposite of what's needed.

Answer:

B. \( y = f(x + 3)-2 \)