To determine the transformations that occur when the original function \( f(x) = \sqrt{x} \) is modified to \( f(x) = -3 \sqrt{x+3} \), we need to carefully analyze each step of the modification.
Step-by-Step Analysis:
1. Multiplying by -3:
- Multiplying the function \( \sqrt{x} \) by \(-3\) results in the function \( -3\sqrt{x} \).
- The multiplication by \(-3\) introduces a factor of \(-1\) and a factor of \(3\).
Transformations caused:
- Reflection over the x-axis: The multiplication by \(-1\) causes the graph to be reflected over the \(x\)-axis, meaning that all positive \(y\)-values will become negative, and all negative \(y\)-values will become positive.
- Vertical dilation: The multiplication by \(3\) causes a vertical stretching (dilation). The \(y\)-values are scaled by a factor of \(3\), making the graph steeper.
2. Replacing \(x\) with \(x + 3\):
- Replacing \(x\) with \(x + 3\) results in the function \( \sqrt{(x+3)} \).
Transformation caused:
- Horizontal translation: The replacement \(x \rightarrow x + 3\) shifts the entire graph horizontally. Specifically, shifting by \(+3\) translates the graph 3 units to the left.
Combining these transformations, we can summarize the effects:
- The graph is reflected over the \(x\)-axis.
- The graph undergoes a vertical dilation (stretching by a factor of 3).
- The graph experiences a horizontal translation (shift to the left by 3 units).
Transformations that do NOT occur:
- There is no reflection over the \(y\)-axis.
- There is no vertical translation (upward or downward shift).
Therefore, the transformations that occur on the new, modified graph \( f(x) = -3 \sqrt{x+3} \) are:
- reflection over the \(x\)-axis
- dilation
- horizontal translation
