Question

reflecting the cube root function quick check
analyze the effect on the y-intercept in the graph of $f(x) = \sqrt3{x}$ when $f(x)$ is replaced by $af(x)$ and $a$ is negative. (1 point)

the y-intercept shifts to the left by a distance of $a$.

the y-intercept shifts down by a distance of $a$.

the y-intercept shifts up by a distance of $a$.

the y-intercept does not change.

Explanation:

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

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

Step2: Find the y-intercept of \( af(x) \)

The new function is \( af(x)=a\sqrt[3]{x} \). Set \( x = 0 \), then \( af(0)=a\times\sqrt[3]{0}=a\times0 = 0 \). So the y-intercept of \( af(x) \) is also \( (0, 0) \), regardless of the value of \( a \) (as long as \( a \) is a real number, including negative numbers). This means the y-intercept does not change when we replace \( f(x) \) with \( af(x) \) (even when \( a \) is negative).

Answer:

The y-intercept does not change.