Question

reflecting the cubic function quick check
if $f(x) = x^3$, what is the effect on its graph when $f(x)$ is replaced by $f(-2x)$? (1 point)
\bigcirc a vertical compression and reflection over the $x$-axis
\bigcirc a vertical stretch and reflection over the $x$-axis
\bigcirc a horizontal compression and reflection over the $y$-axis
\bigcirc a horizontal stretch and reflection over the $y$-axis

Explanation:

Step1: Recall Transformations

For a function \( y = f(x) \), replacing \( x \) with \( -ax \) (where \( a>0 \)) involves horizontal transformations. The general rules:

• Reflection over \( y \)-axis: \( f(-x) \) (replace \( x \) with \( -x \)).
• Horizontal compression/stretch: If \( |a|>1 \), it's a horizontal compression by factor \( \frac{1}{a} \); if \( 0<|a|<1 \), horizontal stretch.

Step2: Analyze \( f(-2x) \)

Given \( f(x)=x^3 \), so \( f(-2x)=(-2x)^3=-8x^3 \). But let's analyze the transformation from \( f(x) \) to \( f(-2x) \) step by step:

1. First, replace \( x \) with \( -x \): \( f(-x)=(-x)^3=-x^3 \) (reflection over \( y \)-axis? Wait, no: \( f(-x) \) is reflection over \( y \)-axis. Wait, actually, when we have \( f(-2x) \), it's equivalent to \( f(-(2x)) \). So first, consider the horizontal compression: replacing \( x \) with \( 2x \) (since \( 2x \) means \( x \) is scaled by 2, so horizontal compression by \( \frac{1}{2} \)), then reflecting over \( y \)-axis (because of the negative sign on \( x \), i.e., \( -2x = -(2x) \)).

Wait, more accurately: For \( y = f(kx) \), if \( |k|>1 \), horizontal compression by \( \frac{1}{|k|} \); if \( k \) is negative, reflection over \( y \)-axis. Here, \( k = -2 \), so \( |k| = 2>1 \), so horizontal compression by \( \frac{1}{2} \), and since \( k \) is negative, reflection over \( y \)-axis.

Now check the options:

• Vertical transformations: These involve multiplying the function by a constant (e.g., \( a f(x) \)), but here we have \( f(-2x) \), which is a horizontal transformation (inside the function, affecting \( x \)), not vertical (which would be outside, like \( -2f(x) \)). So options with vertical compression/stretch are wrong (first two options).
• Now, horizontal: \( |-2| = 2>1 \), so horizontal compression (since \( k = -2 \), replacing \( x \) with \( -2x \) is like \( x \to 2x \) then \( x \to -x \), but the key is \( |k| = 2 \), so horizontal compression by \( \frac{1}{2} \), and reflection over \( y \)-axis (because of the negative sign in \( -2x \), which is \( -(2x) \), so reflection over \( y \)-axis). So the third option: "a horizontal compression and reflection over the y - axis" matches.

Answer:

C. a horizontal compression and reflection over the y - axis (assuming the options are labeled A, B, C, D as: A. a vertical compression and reflection over the x - axis; B. a vertical stretch and reflection over the x - axis; C. a horizontal compression and reflection over the y - axis; D. a horizontal stretch and reflection over the y - axis)