Question

Analyze the effect that −4
has on the graph from g(x)=x−−√+3
to p(x)=−4x−−−−√+3
.(1 point)
Responses

horizontal compression by a factor of 4 and reflection over the x-axis
horizontal compression by a factor of 4 and reflection over the x -axis

horizontal stretch by a factor of 4 and reflection over the x-axis
horizontal stretch by a factor of 4 and reflection over the x -axis

horizontal stretch by a factor of 14
and reflection over the y-axis
horizontal stretch by a factor of Start Fraction 1 over 4 End Fraction and reflection over the y -axis

horizontal compression by a factor of 14
and reflection over the y-axis

Answer 1

To analyze the effect of the transformation from \( g(x) = \sqrt{x} + 3 \) to \( p(x) = -4\sqrt{x} + 3 \), we need to break down the changes step by step.

1. **Reflection over the x-axis:** The negative sign in front of the square root in \( p(x) \) indicates that the graph of \( g(x) \) is reflected over the x-axis. This means that all y-values of the original graph are inverted (multiplied by -1).

2. **Horizontal scaling:** The factor of 4 in front of the square root suggests a horizontal compression. In general, if you have a function \( f(x) \) and you transform it to \( f(kx) \) where \( k > 1 \), it results in a horizontal compression by a factor of \( \frac{1}{k} \). In this case, we have \( p(x) = -4\sqrt{x} + 3 \); it can be viewed as \( f(4x) \). Here, the horizontal compression would be by a factor of \( \frac{1}{4} \).

Based on the analysis:
- The transformation reflects the graph over the x-axis.
- The transformation compresses the graph horizontally by a factor of 4.

Thus, the correct response is:
**horizontal compression by a factor of 4 and reflection over the x-axis.**