https://file.aiquickdraw.com/mathbot/file/1725794095103w8sl1wcy.jpg

The given problem describes a curve with the equation:

\[
x = y^2 - 6y + 11
\]

This curve is rotated **90° clockwise** about a point \( P \), which has coordinates \( (-2, 3) \), to obtain a new curve \( C \). The task is to find the equation of the new curve \( C \).

### Step 1: Equation of the Original Curve
The original curve is:
\[
x = y^2 - 6y + 11
\]
This can also be written in the form \( f(y) \), meaning it relates \( x \) as a function of \( y \).

### Step 2: Shift the Coordinates to the Origin
Since we are rotating around the point \( P(-2, 3) \), we first need to **shift the coordinates** so that \( P \) becomes the origin. To do this, we perform the following translation:
- Let \( x' = x + 2 \)
- Let \( y' = y - 3 \)

Thus, the original curve becomes:
\[
x' - 2 = (y' + 3)^2 - 6(y' + 3) + 11
\]
Simplifying the right-hand side:

\[
(x' - 2) = (y'^2 + 6y' + 9) - 6(y' + 3) + 11
\]
\[
x' - 2 = y'^2 + 6y' + 9 - 6y' - 18 + 11
\]
\[
x' - 2 = y'^2 + 2
\]
\[
x' = y'^2 + 4
\]

So the equation in the shifted coordinates is:
\[
x' = y'^2 + 4
\]

### Step 3: Rotate the Curve 90° Clockwise
To rotate the curve 90° clockwise around the origin, we apply the following transformation:
\[
x'' = y'
\]
\[
y'' = -x'
\]

Thus, the new curve in the rotated coordinates is:
\[
y'' = -(y'^2 + 4)
\]
or equivalently:
\[
y'' = -y'^2 - 4
\]

### Step 4: Shift Back to the Original Coordinates
Now, we reverse the shift to return to the original coordinates \( x \) and \( y \):
\[
x = x'' + 2
\]
\[
y = y'' + 3
\]

Substitute \( y'' = -y'^2 - 4 \) into the second equation:
\[
y = -y'^2 - 4 + 3
\]
\[
y = -y'^2 - 1
\]

Thus, the final equation of the new curve \( C \) is:
\[
y = -(x - 2)^2 - 1
\]

This is the equation of the curve \( C \) after a 90° clockwise rotation about the point \( P(-2, 3) \).

Would you like further clarification or details?

---

Here are 5 related questions for practice:

1. How would the equation change if the rotation was 90° counterclockwise?
2. What is the general transformation matrix for rotating a point around an arbitrary point?
3. How would you solve for the new equation if the curve was translated instead of rotated?
4. What are the key differences in the equations of curves when rotating by different angles (e.g., 180°, 270°)?
5. How do you find the points of intersection between two rotated curves?

**Tip**: Always break down rotations into translations to the origin and apply coordinate transformations step by step to avoid mistakes.

The curve with equation x = y^2 - 6y + 11 is rotated 90° clockwise about the point P (-2, 3) to give the curve C. What is the equation of curve C?

Learn how to find the new equation of a curve after a 90° clockwise rotation around a point. In this problem, the curve x = y^2 - 6y + 11 is rotated around point P(-2, 3), illustrating key concepts in coordinate geometry and transformations. Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation