Question

analyze the effect on the graph when $f(x)=x^3$ is replaced with $f(x)=(bx)^3$ where $b > 1$. which option correctly describes the effect? (1 point)

• the graph is shifted to the right
• the graph is horizontally stretched
• the graph is shifted to the left
• the graph is horizontally compressed

graphing calculator

Explanation:

Step1: Recall Horizontal Transformations

For a function \( y = f(kx) \), when \( |k| > 1 \), it's a horizontal compression; when \( 0 < |k| < 1 \), it's a horizontal stretch. Here, the original function is \( f(x)=x^3 \), and the new function is \( f(x)=(bx)^3=f(bx) \) with \( b > 1 \).

Step2: Analyze the Transformation

Since \( b > 1 \), comparing to the rule of horizontal transformations (\( y = f(kx) \), \( k = b>1 \)), this means the graph of \( f(x)=(bx)^3 \) is a horizontal compression of the graph of \( f(x)=x^3 \). Shifts (left/right) involve adding/subtracting inside the function, not multiplying the input by a constant greater than 1. A horizontal stretch would occur if \( 0 < b < 1 \), but here \( b > 1 \), so it's a compression.

Answer:

the graph is horizontally compressed