Find Abc Quartic E08Bf7

1. **State the problem:** We are given a quartic function of the form $$y = a(x - b)^2(x - c)^2$$ and a graph with x-intercepts at $(-2,0)$ and $(1,0)$, and a point $(0,2)$ on the curve. 2. **Identify the roots:** Since the graph touches the x-axis at $x = -2$ and $x = 1$ and does not cross it, these are roots with multiplicity 2. Thus, $b = -2$ and $c = 1$. 3. **Write the function with known roots:** $$y = a(x - (-2))^2 (x - 1)^2 = a(x + 2)^2 (x - 1)^2$$ 4. **Use the given point to find $a$:** Substitute $x=0$ and $y=2$ into the equation: $$2 = a(0 + 2)^2 (0 - 1)^2 = a(2)^2 ( -1)^2 = a \times 4 \times 1 = 4a$$ 5. **Solve for $a$:** $$4a = 2 \implies a = \frac{2}{4} = \frac{1}{2}$$ 6. **Final equation:** $$y = \frac{1}{2} (x + 2)^2 (x - 1)^2$$ **Answer:** $a = \frac{1}{2}$, $b = -2$, $c = 1$.