Quadratic Equation 612C85

1. **State the problem:** We need to find the equation of a quadratic function in standard form $$y = ax^2 + bx + c$$ given points on the graph. 2. **Identify known points:** The vertex is at $$(-1.5, 6.75)$$ and the parabola passes through $$(-3, 6)$$. 3. **Use vertex form:** The vertex form of a quadratic is $$y = a(x - h)^2 + k$$ where $$(h, k)$$ is the vertex. Here, $$h = -1.5$$ and $$k = 6.75$$, so: $$y = a(x + 1.5)^2 + 6.75$$ 4. **Use the point $$(-3, 6)$$ to find $$a$$:** Substitute $$x = -3$$ and $$y = 6$$: $$6 = a(-3 + 1.5)^2 + 6.75$$ $$6 = a(-1.5)^2 + 6.75$$ $$6 = a(2.25) + 6.75$$ 5. **Solve for $$a$$:** $$6 - 6.75 = 2.25a$$ $$-0.75 = 2.25a$$ $$a = \frac{-0.75}{2.25}$$ $$a = \frac{\cancel{-0.75}}{\cancel{2.25}} = -\frac{1}{3}$$ 6. **Write the vertex form with $$a$$:** $$y = -\frac{1}{3}(x + 1.5)^2 + 6.75$$ 7. **Convert to standard form:** Expand the square: $$(x + 1.5)^2 = x^2 + 3x + 2.25$$ Substitute: $$y = -\frac{1}{3}(x^2 + 3x + 2.25) + 6.75$$ Distribute $$-\frac{1}{3}$$: $$y = -\frac{1}{3}x^2 - x - 0.75 + 6.75$$ Combine constants: $$y = -\frac{1}{3}x^2 - x + 6$$ **Final answer:** $$\boxed{y = -\frac{1}{3}x^2 - x + 6}$$