Quadratic Coefficients 393756

1. **State the problem:** We need to find the coefficients $a$, $b$, and $c$ of the quadratic equation $y = ax^2 + bx + c$ given the graph. 2. **Analyze the graph:** The parabola opens upward and has its vertex at approximately $(4, 0)$. 3. **Use vertex form:** The vertex form of a quadratic is $$y = a(x - h)^2 + k$$ where $(h, k)$ is the vertex. 4. **Substitute vertex:** Here, $h = 4$ and $k = 0$, so $$y = a(x - 4)^2 + 0 = a(x - 4)^2$$ 5. **Expand vertex form:** Expanding gives $$y = a(x^2 - 8x + 16) = a x^2 - 8a x + 16a$$ 6. **Match standard form:** Comparing to $y = ax^2 + bx + c$, we identify: $$a = a$$ $$b = -8a$$ $$c = 16a$$ 7. **Find $a$ using a point:** Since the vertex is at $(4,0)$, the parabola passes through this point. To find $a$, we need another point on the graph. Since the problem does not provide another point, we cannot determine the exact value of $a$, $b$, and $c$ numerically. 8. **Conclusion:** The coefficients are related by $$b = -8a$$ and $$c = 16a$$ with $a > 0$ because the parabola opens upward. **Final answer:** $$a = a$$ $$b = -8a$$ $$c = 16a$$ where $a$ is a positive constant.