Question

reflecting the cubic function quick check
analyze the effect on the graph of $f(x) = x^3$ when $f(x)$ is replaced by $f(bx)$ for $b = -1$. (1 point)
\bigcirc a reflection about the $y$-axis
\bigcirc a reflection about the line $y = x$
\bigcirc a shift to the right
\bigcirc a shift to the left

Explanation:

To determine the effect of replacing \( f(x) = x^3 \) with \( f(bx) \) when \( b = -1 \), we substitute \( b = -1 \) into \( f(bx) \). So \( f(bx) = f(-x) = (-x)^3=-x^3\).

For a function \( y = f(x) \), replacing \( x \) with \( -x \) gives \( y = f(-x) \), which is a reflection about the \( y \)-axis.

• A reflection about the line \( y = x \) would involve finding the inverse function, which is not the case here.
• Shifts to the right or left involve replacing \( x \) with \( x - h \) (shift right) or \( x+h \) (shift left) for some \( h

eq0 \), not multiplying \( x \) by a constant.

So the transformation \( f(x)\to f(-x) \) is a reflection about the \( y \)-axis.

Answer:

A. a reflection about the \( y \)-axis