Question

Analyze the effect on the y-intercept in the graph of f(x)=x−−√3
when f(x)
is replaced by f(bx)
and b is less than −1
.(1 point)
Responses

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

The y-intercept stretches horizontally.
The y -intercept stretches horizontally.

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

The y-intercept compresses horizontally.

Answer 1

To analyze how the function \( f(x) = \sqrt{x} - 3 \) behaves when it is replaced by \( f(bx) \) where \( b < -1 \), we can follow these steps:

1. **Determine the original y-intercept:** The original function \( f(x) \) has a y-intercept found by evaluating the function at \( x = 0 \):
\[
f(0) = \sqrt{0} - 3 = -3
\]

2. **Substitute \( x \) with \( bx \):** Now, we substitute \( x \) with \( bx \) in the function to get \( f(bx) = \sqrt{bx} - 3 \).

3. **Find the new y-intercept:** To find the y-intercept of the transformed function, we evaluate it at \( bx = 0 \) (which implies \( x = 0 \)):
\[
f(b \cdot 0) = \sqrt{b \cdot 0} - 3 = -3
\]

4. **Analyze the effect of \( b < -1 \):** The transformation \( f(bx) \) does not change the y-intercept, which remains at \(-3\). The negative value and magnitude of \( b \) affect the horizontal scaling and reflection of the graph, but they do not influence the y-intercept.

Based on the above analysis, the correct response is:

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