Question

Analyze the effect on the y-intercept in the graph of f(x)=x−−√3 when f(x) is replaced by af(x) and a is negative.(1 point) Responses The y-intercept does not change. The y -intercept does not change. The y-intercept shifts down by a distance of a. The y -intercept shifts down by a distance of a. The y-intercept shifts up by a distance of a. The y -intercept shifts up by a distance of a . The y-intercept shifts to the left by a distance of a.

Answer 1

To analyze the effect on the y-intercept when \( f(x) \) is replaced by \( af(x) \) with \( a \) being negative, we first need to determine the original y-intercept of \( f(x) = \sqrt{x} - 3 \).

1. **Finding the original y-intercept:**
To find the y-intercept, we substitute \( x = 0 \) into the function:
\[
f(0) = \sqrt{0} - 3 = -3.
\]
Therefore, the original y-intercept is at the point \((0, -3)\).

2. **Transforming the function:**
Now consider the transformed function \( af(x) \):
\[
af(x) = a(\sqrt{x} - 3).
\]
The new y-intercept can be found by substituting \( x = 0 \) again:
\[
af(0) = a(\sqrt{0} - 3) = a(-3) = -3a.
\]

3. **Effect of a being negative:**
Since \( a \) is negative, \( -3a \) will be a positive value when \( a \) is negative. This means the y-intercept moves from \(-3\) to a positive value that is further up the y-axis.

4. **Conclusion:**
The y-intercept shifts **up** by a distance of **|a| \cdot 3** (which depends on the magnitude of \( a \)).

Among the given responses, the most accurate description (considering any generic negative \( a \)) is:
- **The y-intercept shifts up by a distance of \( |3a| \) (where \( |a| \) is positive and the idea of shifting down can be interpreted based on the negative effect of \( a \))**.

However, in terms of the options provided, the precise phrases might not all completely reflect the shifting caused by \( a \) without specifics on numerical values. Since \( a < 0 \) leads to some ambiguity in the provided options, a more generalized conclusion would be:
- **The y-intercept shifts up (taking into account the magnitude of the negative value of \( a \)).**

In summary, the key takeaway is that the y-intercept indeed shifts depending on the behavior of \( a \), specifically going to a positive intercept from a negative one as the original function shifts based on negative scaling.