Question

What is the effect on the y-intercept in the graph of f(x)=x−−√3
when f(x)
is replaced by −5f(x)
?(1 point)
Responses

The y-intercept does not change.
The y -intercept does not change.

The y-intercept shifts to the right by a distance of 5.
The y -intercept shifts to the right by a distance of 5.

The y-intercept shifts up by a distance of 5.
The y -intercept shifts up by a distance of 5.

The y-intercept shifts down by a distance of 5.

Answer 1

To determine the effect on the y-intercept when the function \( f(x) = \sqrt{x} - 3 \) is replaced by \( -5f(x) \), we first need to calculate the original y-intercept of \( f(x) \).

The y-intercept occurs when \( x = 0 \). For \( f(x) \):

\[
f(0) = \sqrt{0} - 3 = -3
\]

Thus, the y-intercept of \( f(x) \) is \(-3\).

Now, let's find the y-intercept for the transformed function:

\[
-5f(x) = -5(\sqrt{x} - 3)
\]

When we calculate the y-intercept of \( -5f(x) \), we again set \( x = 0 \):

\[
-5f(0) = -5(-3) = 15
\]

So, the y-intercept of \( -5f(x) \) is \( 15 \).

Now we compare the y-intercepts:

- Original y-intercept: \(-3\)
- New y-intercept: \(15\)

To find the change:

\[
15 - (-3) = 15 + 3 = 18
\]

Thus, the y-intercept shifts **up** by a distance of **18**.

Given the options provided, it looks like none of them state an upward shift by 18. However, from the context, you could take the concept of transformation into account.

The closest effect to note is that the function has undergone a vertical transformation. Since you originally had the function shift downwards to -3 and now it has moved upwards to 15, it means it did indeed shift upwards.

However, since the details of how much the shift occurred doesn't directly correspond with any provided options regarding distance 5, the conclusion based on given options would be:

None of the provided options accurately reflect the true transformation that occurred based on the output values. The value strictly concludes a dramatic upward shift rather than downwards.

But to summarize based on the options, **none are correct. However if asked generally, we note a upward shift.**