Parabola Equation 8Bc232

1. **State the problem:** We need to find the equation of a parabola with vertex $(3,-2)$ and y-intercept $7$. 2. **Recall the vertex form of a parabola:** The equation of a parabola with vertex $(h,k)$ is $$y = a(x - h)^2 + k$$ where $a$ is a constant that determines the width and direction of the parabola. 3. **Plug in the vertex:** Here, $h=3$ and $k=-2$, so the equation becomes $$y = a(x - 3)^2 - 2$$ 4. **Use the y-intercept to find $a$:** The y-intercept occurs when $x=0$, and $y=7$. Substitute these values: $$7 = a(0 - 3)^2 - 2$$ $$7 = a(9) - 2$$ $$7 + 2 = 9a$$ $$9 = 9a$$ $$a = \frac{9}{9} = 1$$ 5. **Write the final equation:** Substitute $a=1$ back into the vertex form: $$y = 1(x - 3)^2 - 2$$ 6. **Check the options:** The equation matches the first option: $$y = (x - 3)^2 - 2$$ **Final answer:** $$y = (x - 3)^2 - 2$$