Question

Rotations as Functions Quick Check 4 of 54 of 5 Items Question A vertex of △PQR is rotated from Q(−4,−9) to Q′(−9,4). Use rotation function mapping to determine where the image of the other two vertices P(−4,0) and R(4,−4), should be placed.(1 point) Responses P′(0,4) and R′(−4,4) upper P prime left parenthesis 0 comma 4 right parenthesis and upper R left parenthesis negative 4 comma 4 right parenthesis P′(0,4) and R′(−4,−4) upper P prime left parenthesis 0 comma 4 right parenthesis and upper R left parenthesis negative 4 comma negative 4 right parenthesis P′(0,−4) and R′(−4,4) upper P prime left parenthesis 0 comma negative 4 right parenthesis and upper R left parenthesis negative 4 comma 4 right parenthesis P′(0,−4) and R′(−4,−4) upper P prime left parenthesis 0 comma negative 4 right parenthesis and upper R left parenthesis negative 4 comma negative 4 right parenthesis Skip to navigation page 15 of 15 Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.

Answer 1

To determine the images of the other vertices \(P\) and \(R\) after the rotation that maps \(Q(-4, -9)\) to \(Q'(-9, 4)\), we need to establish the rotation function that has been applied.

### Step 1: Determine the Rotation Angle and Center

The point \(Q(-4, -9)\) is rotated to \(Q'(-9, 4)\). We can analyze the movement:
- The transformation includes a significant change in both x and y coordinates, suggesting a rotation.

To find the angle of rotation, we can compute how the point rotates:
1. **From** \((-4, -9)\) to \((-9, 4)\)
- The original point \(Q\) can be translated to the origin, then rotated, and translated back.

### Step 2: Check the Rotation Function
Based on the transformation observed, it suggests a 90-degree counterclockwise rotation. The rotation matrix for a 90-degree counterclockwise rotation is given by:

\[
\left( \begin{array}{cc}
0 & -1 \\
1 & 0 \\
\end{array} \right)
\]

### Step 3: Apply the Rotation to Other Points
We now apply this rotation to the points \(P(-4, 0)\) and \(R(4, -4)\).

#### For Point P:
\[
P(-4, 0) \rightarrow P'(0, 4)
\]

#### For Point R:
\[
\begin{align*}
R(4, -4) &\rightarrow R'(-(-4), 4) \\
R' & = (-(-4), 4) \\
R' & = (4, 4)
\end{align*}
\]

### Conclusion
Thus, the images after the rotation function are:
- \(P' = (0, 4)\)
- \(R' = (4, 4)\)

Given your response options, it appears there are no matches. However, based on the calculations:

- **From the options given, the closest results for \(P'\) and \(R'\) do not align with the answer based on our calculations.**

It seems to indicate that there might be a miscommunication or a misunderstanding of the specific prompt about the rotation center (which could significantly change outcomes). Please double-check to confirm specific angles or constraints if available in your task outline concerning rotation axis or angle. Thus the results are noted as:
- \(P' = (0, 4)\)
- \(R' = (4, 4)\) or matching where we can align better outputs based upon what's presented.