1. **Problem Statement:** Given graphs of parabolas, write expressions for the parabolas that match the graphs where pairs of parabolas are identical. 2. **General form of a parabola:** A parabola opening upwards or downwards can be expressed as: $$y = a(x - h)^2 + k$$ where $(h,k)$ is the vertex and $a$ controls the width and direction (positive $a$ opens upwards, negative $a$ opens downwards). 3. **Important rules:** - Parabolas that are identical have the same $a$, $h$, and $k$. - Shifts left/right change $h$. - Shifts up/down change $k$. 4. **From the first graph (top-left):** - Orange parabola $g(x)$ opens upwards and is shifted left. - Black parabola $h(x)$ opens upwards and is shifted right. Assuming the vertex of $g$ is at $(-2,0)$ and $h$ at $(2,0)$ with $a=1$ (standard width): $$g(x) = (x + 2)^2$$ $$h(x) = (x - 2)^2$$ 5. **From the second graph (center-left):** - Purple parabola $f(x)$ opens upwards on the left. - Orange parabola $g(x)$ opens upwards on the right. Assuming $f$ vertex at $(-3,0)$ and $g$ vertex at $(1,0)$ with $a=1$: $$f(x) = (x + 3)^2$$ $$g(x) = (x - 1)^2$$ 6. **From the third graph (bottom-left):** - Purple parabola $f(x)$ opens upwards on the left. - Green parabola $p(x)$ opens upwards on the right. Assuming $f$ vertex at $(-3,0)$ and $p$ vertex at $(3,0)$ with $a=1$: $$f(x) = (x + 3)^2$$ $$p(x) = (x - 3)^2$$ **Final answers:** $$g(x) = (x + 2)^2$$ $$h(x) = (x - 2)^2$$ $$f(x) = (x + 3)^2$$ $$g(x) = (x - 1)^2$$ $$f(x) = (x + 3)^2$$ $$p(x) = (x - 3)^2$$