Question

reflecting the cube root function quick check
what is the effect on the y-intercept in the graph of $f(x)=\sqrt3{x}$ when $f(x)$ is replaced by $-5f(x)$? (1 point)
\bigcirc the y-intercept does not change.
\bigcirc the y-intercept shifts down by a distance of 5.
\bigcirc the y-intercept shifts to the right by a distance of 5.
\bigcirc the y-intercept shifts up by a distance of 5.

Explanation:

Step1: Find y-intercept of \( f(x) = \sqrt[3]{x} \)

To find the y-intercept, set \( x = 0 \). So \( f(0) = \sqrt[3]{0} = 0 \). The y-intercept is \( (0, 0) \).

Step2: Find y-intercept of \( -5f(x) \)

First, substitute \( f(x) \) with \( \sqrt[3]{x} \), so \( -5f(x) = -5\sqrt[3]{x} \). Now find the y-intercept by setting \( x = 0 \): \( -5\sqrt[3]{0} = -5 \times 0 = 0 \). Wait, no, wait. Wait, no, let's recalculate. Wait, \( -5f(x) = -5\sqrt[3]{x} \). When \( x = 0 \), \( -5\sqrt[3]{0} = 0 \)? Wait, that can't be right. Wait, no, maybe I made a mistake. Wait, original function \( f(x) = \sqrt[3]{x} \), y-intercept at \( x=0 \), \( f(0)=0 \). Now the new function is \( g(x) = -5f(x) = -5\sqrt[3]{x} \). Then \( g(0) = -5\sqrt[3]{0} = 0 \)? Wait, that's the same? But that contradicts the options. Wait, no, maybe I messed up the transformation. Wait, no, let's check again. Wait, the original function: \( f(x) = \sqrt[3]{x} \), y-intercept (0,0). The new function is \( -5f(x) \), so when \( x=0 \), \( -5 \times 0 = 0 \). Wait, but the options are about shifting. Wait, maybe I made a mistake. Wait, no, the y-intercept is the value when \( x=0 \). So for \( f(x) \), it's 0. For \( -5f(x) \), it's also 0? But that would mean the y-intercept does not change. But let's check the options. The first option is "The y-intercept does not change." But wait, maybe I made a mistake in the transformation. Wait, no, let's think again. The y-intercept is at \( x=0 \). So for \( f(x) \), \( f(0) = 0 \). For \( -5f(x) \), when \( x=0 \), \( -5f(0) = -5 \times 0 = 0 \). So the y-intercept is still \( (0, 0) \). So the y-intercept does not change. Wait, but the options include that. So the correct answer is the first option. Wait, but let me check again. Wait, maybe I misread the problem. The problem says "when \( f(x) \) is replaced by \( -5f(x) \)". So the new function is \( y = -5\sqrt[3]{x} \). The y-intercept is when \( x=0 \), so \( y = -5\sqrt[3]{0} = 0 \). So the y-intercept is still (0,0), same as the original. So the y-intercept does not change.

Answer:

The y-intercept does not change. (The first option: "The y-intercept does not change.")