1. **Problem:** Find the new equation of the line after a 90° clockwise rotation about the point (-2, 3) for the line $y=2x+2$. 2. **Formula and rules:** - Rotation of a point $(x,y)$ about a point $(h,k)$ by 90° clockwise is given by: $$x' = h + (y - k)$$ $$y' = k - (x - h)$$ - To find the new line equation, transform two points on the original line and find the line through their images. 3. **Intermediate work:** - Choose two points on $y=2x+2$: - When $x=0$, $y=2(0)+2=2$, so point $P_1=(0,2)$ - When $x=1$, $y=2(1)+2=4$, so point $P_2=(1,4)$ - Rotate $P_1$ about $(-2,3)$: $$x'_1 = -2 + (2 - 3) = -2 - 1 = -3$$ $$y'_1 = 3 - (0 + 2) = 3 - 2 = 1$$ So $P'_1 = (-3,1)$ - Rotate $P_2$ about $(-2,3)$: $$x'_2 = -2 + (4 - 3) = -2 + 1 = -1$$ $$y'_2 = 3 - (1 + 2) = 3 - 3 = 0$$ So $P'_2 = (-1,0)$ - Find the line through $P'_1(-3,1)$ and $P'_2(-1,0)$: Slope $m = \frac{0 - 1}{-1 + 3} = \frac{-1}{2} = -\frac{1}{2}$ - Equation using point-slope form: $$y - 1 = -\frac{1}{2}(x + 3)$$ $$y = -\frac{1}{2}x - \frac{3}{2} + 1 = -\frac{1}{2}x - \frac{1}{2}$$ 4. **Answer:** The new line equation after rotation is $$y = -\frac{1}{2}x - \frac{1}{2}$$