Question

describe the transformation from ( v(x) = x^3 + 2 ) to ( q(x) = (0.25x)^3 + 2 ). (1 point)

• horizontal compression by a factor of 4
• horizontal stretch by a factor of 4
• horizontal compression by a factor of 0.25
• horizontal stretch by a factor of 0.25

graphing calculator

Explanation:

Step1: Recall Horizontal Stretch/Compression Rule

For a function \( y = f(kx) \), if \( |k| < 1 \), it's a horizontal stretch by factor \( \frac{1}{|k|} \); if \( |k| > 1 \), horizontal compression by \( \frac{1}{|k|} \). Here, \( v(x)=x^3 + 2 \), \( q(x)=(0.25x)^3+2=f(0.25x) \) where \( f(x)=x^3 + 2 \), \( k = 0.25=\frac{1}{4} \).

Step2: Determine Transformation Type

Since \( k=\frac{1}{4} \) ( \( |k|=\frac{1}{4}<1 \) ), horizontal stretch factor is \( \frac{1}{k}=\frac{1}{\frac{1}{4}} = 4 \). So it's a horizontal stretch by factor 4.

Answer:

B. horizontal stretch by a factor of 4