Question
Analyze the effect on the \( y \)-intercept in the graph of \( f(x)=\sqrt[3]{x} \) when \( f(x) \) is replaced by \( f(b x) \) and \( b \) is less than -1. (1 point) The \( y \)-intercept shifts up by a distance of \( b \). The \( y \)-intercept stretches horizontally. The \( y \)-intercept does not change. The \( y \)-intercept compresses horizontally.
Ask by Hodges Bryant. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The \( y \)-intercept does not change.
Solution
When analyzing the transformation \( f(bx) = \sqrt[3]{bx} \) for \( b < -1 \), we focus on the \( y \)-intercept. The \( y \)-intercept occurs where \( x = 0 \).
\[
f(b \cdot 0) = \sqrt[3]{0} = 0
\]
Regardless of the value of \( b \), as long as \( x = 0 \), the output is always 0. Therefore, the \( y \)-intercept remains unchanged.
**Answer:** The \( y \)-intercept does not change.
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Simplify this solution The Deep Dive
When we replace \( f(x) \) with \( f(bx) \) for \( b < -1 \), we are effectively analyzing the new function \( f(bx) = \sqrt[3]{bx} \). Since the \( y \)-intercept is determined by the function value at \( x = 0 \), we calculate \( f(b \cdot 0) = f(0) = \sqrt[3]{0} \), which still equals 0. Thus, the \( y \)-intercept does not change; it's still at the point (0,0) regardless of the value of \( b \). So, the correct answer is: The \( y \)-intercept does not change.
