1. **State the problem:** Write the quadratic function in vertex form $f(x) = a(x-p)^2 + q$ for each graph. 2. **Recall vertex form:** The vertex form of a quadratic is $$f(x) = a(x-p)^2 + q$$ where $(p,q)$ is the vertex and $a$ determines the opening direction and width. --- ### Part (a): 3. **Identify vertex and points:** Vertex: $(-4, 2)$ Points: $(-6, 10)$ and $(0, 10)$ 4. **Plug vertex into form:** $$f(x) = a(x + 4)^2 + 2$$ 5. **Use point $(-6, 10)$ to find $a$:** $$10 = a(-6 + 4)^2 + 2$$ $$10 = a(-2)^2 + 2$$ $$10 = 4a + 2$$ 6. **Solve for $a$:** $$10 - 2 = 4a$$ $$8 = 4a$$ $$a = \frac{8}{4}$$ $$a = 2$$ 7. **Write final function for (a):** $$f(x) = 2(x + 4)^2 + 2$$ --- ### Part (b): 8. **Identify vertex and points:** Vertex: $(0, 4)$ Points: $(-2, 2)$ and $(2, 2)$ 9. **Plug vertex into form:** $$f(x) = a(x - 0)^2 + 4 = a x^2 + 4$$ 10. **Use point $(-2, 2)$ to find $a$:** $$2 = a(-2)^2 + 4$$ $$2 = 4a + 4$$ 11. **Solve for $a$:** $$2 - 4 = 4a$$ $$-2 = 4a$$ $$a = \frac{-2}{4}$$ $$a = -\frac{1}{2}$$ 12. **Write final function for (b):** $$f(x) = -\frac{1}{2} x^2 + 4$$