1. **Problem statement:** Given the parabola $$y = -3(x + 1.5)^2 + 1$$, find the equations of the parabola after reflection about the colored axes: the blue axis $$y = -1$$ and the green axis $$x = -1$$. 2. **Reflection rules:** - Reflection about a horizontal line $$y = k$$ transforms a point $$(x,y)$$ to $$(x, 2k - y)$$. - Reflection about a vertical line $$x = h$$ transforms a point $$(x,y)$$ to $$(2h - x, y)$$. 3. **Original function:** $$y = -3(x + 1.5)^2 + 1$$. --- ### a) Reflection about the blue axis $$y = -1$$ - The reflection formula for $$y$$ is $$y' = 2(-1) - y = -2 - y$$. - Substitute $$y$$ from the original function: $$y' = -2 - \left[-3(x + 1.5)^2 + 1\right] = -2 + 3(x + 1.5)^2 - 1 = 3(x + 1.5)^2 - 3$$ - So the reflected function is: $$\boxed{y = 3(x + 1.5)^2 - 3}$$ --- ### b) Reflection about the green axis $$x = -1$$ - The reflection formula for $$x$$ is $$x' = 2(-1) - x = -2 - x$$. - Substitute $$x'$$ into the original function: $$y = -3\left((-2 - x) + 1.5\right)^2 + 1 = -3(-0.5 - x)^2 + 1$$ - Simplify inside the square: $$-0.5 - x = -(x + 0.5)$$ - Since squaring removes the negative sign: $$y = -3(x + 0.5)^2 + 1$$ - So the reflected function is: $$\boxed{y = -3(x + 0.5)^2 + 1}$$ --- ### c) Reflection about both blue and green axes - First reflect about green axis (as above): $$y = -3(x + 0.5)^2 + 1$$ - Then reflect about blue axis $$y = -1$$: $$y' = -2 - y = -2 - \left[-3(x + 0.5)^2 + 1\right] = -2 + 3(x + 0.5)^2 - 1 = 3(x + 0.5)^2 - 3$$ - So the function after both reflections is: $$\boxed{y = 3(x + 0.5)^2 - 3}$$