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 greater than 1.(1 point)
Responses

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

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

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

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

Answer 1

To analyze the effect on the y-intercept when the function \( f(x) = \sqrt{x} - 3 \) is replaced by \( f(bx) \) where \( b > 1 \), we first need to determine what happens to the y-intercept in this transformation.

1. **Finding the y-intercept of the original function**:
- The y-intercept occurs when \( x = 0 \). Therefore, \( f(0) = \sqrt{0} - 3 = -3 \).
- So, the y-intercept of the original function \( f(x) \) is at (0, -3).

2. **Finding the y-intercept of the transformed function**:
- Now, consider the transformed function \( f(bx) \):
\[
f(bx) = \sqrt{bx} - 3
\]
- To find the new y-intercept, we set \( x = 0 \):
\[
f(b \cdot 0) = \sqrt{b \cdot 0} - 3 = \sqrt{0} - 3 = -3
\]
- The y-intercept of the transformed function \( f(bx) \) is still at (0, -3).

**Conclusion**:
Since the y-intercept remains unchanged at (-3) when replacing \( f(x) \) with \( f(bx) \), we conclude that:

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