Question

scaling 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 positive. (1 point)

• the y-intercept shifts to the left by a distance of $a$.
• the y-intercept does not change.
• the y-intercept shifts up by a distance of $a$.
• the y-intercept shifts down by a distance of $a$.

Explanation:

Step1: Find 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 y-intercept of \( af(x) \)

Now consider \( af(x)=a\sqrt[3]{x} \). Set \( x = 0 \), then \( af(0)=a\sqrt[3]{0}=a\times0 = 0 \). So the y-intercept of \( af(x) \) is also \( (0, 0) \).

Step3: Compare y-intercepts

Since both \( f(x) \) and \( af(x) \) (with \( a>0 \)) have the y-intercept at \( (0, 0) \), the y-intercept does not change.

Answer:

The y-intercept does not change.