Question

suppose that $f(x) = x^3$ and $g(x) = -3x^3$. which statement best compares the graph of $g(x)$ with the graph of $f(x)$?
a. the graph of $g(x)$ is the graph of $f(x)$ flipped over the $y$-axis and stretched vertically
b. the graph of $g(x)$ is the graph of $f(x)$ flipped over the $y$-axis and compressed vertically
c. the graph of $g(x)$ is the graph of $f(x)$ flipped over the $x$-axis and compressed vertically
d. the graph of $g(x)$ is the graph of $f(x)$ flipped over the $x$-axis and stretched vertically

Explanation:

1. Recall the transformations of functions: For a function \( y = a\cdot f(x) \), if \( a < 0 \), the graph is flipped over the \( x \)-axis (reflection over \( x \)-axis). If \( |a|>1 \), the graph is vertically stretched; if \( 0<|a|<1 \), it is vertically compressed.
2. For \( F(x)=x^{3} \) and \( G(x) = - 3x^{3}\), here \( a=-3 \). The negative sign means reflection over the \( x \)-axis. Since \( | - 3|=3>1 \), the graph is vertically stretched.
3. Option A and B talk about flipping over \( y \)-axis, which is incorrect (flipping over \( y \)-axis is for \( y = f(-x) \), not \( y=-af(x) \)). Option C says compressed vertically, but \( | - 3|>1 \) means stretched, not compressed. So Option D is correct.

Answer:

D. The graph of \( G(x) \) is the graph of \( F(x) \) flipped over the \( x \)-axis and stretched vertically