Question
What is the effect on the \( y \)-intercept in the graph of \( f(x)=\sqrt[3]{x} \) when \( f(x) \) is replaced by \( -5 f(x) \) ? (1 point) The \( y \)-intercept shifts to the right by a distance of 5 . The \( y \)-intercept shifts down by a distance of 5 . The \( y \)-intercept does not change. The \( y \)-intercept shifts up by a distance of 5 .
Ask by Huff Norris. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The \( y \)-intercept does not change.
Solution
The \( y \)-intercept of a function is determined by the value of the function when \( x = 0 \).
For the original function \( f(x) = \sqrt[3]{x} \):
\[
f(0) = \sqrt[3]{0} = 0
\]
When replacing \( f(x) \) with \( -5f(x) \):
\[
-5f(0) = -5 \times 0 = 0
\]
Since the \( y \)-intercept remains \( 0 \) after the transformation, it does not change.
**Answer:**
The \( y \)-intercept does not change.
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Simplify this solution Extra Insights
When we replace \( f(x) \) with \( -5 f(x) \), we effectively multiply the output of the function by -5. The original function \( f(x) = \sqrt[3]{x} \) has a \( y \)-intercept at point (0, 0). After the transformation, the new function becomes \( g(x) = -5\sqrt[3]{x} \). For \( x = 0 \), \( g(0) = -5\sqrt[3]{0} = -5 \times 0 = 0 \). So, the \( y \)-intercept at (0, 0) remains unchanged even though the output of all y-values (except at the intercept) is scaled and inverted. The answer is: The \( y \)-intercept does not change.
