Question
What is the effect on the \( y \)-intercept in the graph of \( f(x)=\sqrt[3]{x} \) when \( f(x) \) is replaced by \( f(x)+5 \) ? (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 shifts to the left by a distance of 5 . The \( y \)-intercept shifts up by a distance of 5 .
Ask by Reyes Reyes. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The \( y \)-intercept shifts up by a distance of 5.
Solution
When you replace \( f(x) = \sqrt[3]{x} \) with \( f(x) + 5 \), the new function becomes \( \sqrt[3]{x} + 5 \).
- **Original y-intercept:** For \( f(x) = \sqrt[3]{x} \), the y-intercept is at \( (0, 0) \).
- **New y-intercept:** For \( f(x) + 5 = \sqrt[3]{x} + 5 \), when \( x = 0 \), the y-intercept is \( 0 + 5 = 5 \).
This means the y-intercept moves from \( (0, 0) \) to \( (0, 5) \), which is a shift **upwards** by 5 units.
**Answer:** The \( y \)-intercept shifts up by a distance of 5.
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Simplify this solution Beyond the Answer
When we replace \( f(x) \) with \( f(x) + 5 \), we are essentially shifting the entire graph of the function vertically. The original function \( f(x) = \sqrt[3]{x} \) intersects the \( y \)-axis at the point \( (0,0) \). After adding 5, the new function \( f(x) + 5 = \sqrt[3]{x} + 5 \) will intersect the \( y \)-axis at the point \( (0, 5) \). This means the \( y \)-intercept shifts up by a distance of 5. So, the correct answer is: The \( y \)-intercept shifts up by a distance of 5.
