Question

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

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

The y-intercept stretches horizontally by a factor of 5.
The y -intercept stretches horizontally by a factor of 5.

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

The y-intercept compresses horizontally by a factor of 5.

Answer 1

To analyze the effect of replacing \( f(x) \) with \( f(5x) \) in the function \( f(x) = \sqrt{x} + 3 \), we need to look at how this transformation affects the y-intercept.

1. The original function \( f(x) = \sqrt{x} + 3 \) will have a y-intercept when \( x = 0 \):
\[
f(0) = \sqrt{0} + 3 = 3.
\]
So the original y-intercept is at (0, 3).

2. Now, if we replace \( f(x) \) with \( f(5x) \), we have:
\[
f(5x) = \sqrt{5x} + 3.
\]
We find the new y-intercept by substituting \( x = 0 \):
\[
f(5(0)) = f(0) = \sqrt{0} + 3 = 3.
\]

The y-intercept remains at \( (0, 3) \), meaning it does not change at all. Hence, the correct response is:

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