We are given three points $(1, 2)$, $(2, 6)$, and $(-1, 18)$ and need to find the coefficients $a$, $b$, and $c$ in the general form of the parabola:

$y = ax^2 + bx + c$

Substitute each point into the equation to form a system of three equations.

1. For $(1, 2)$:

$2 = a(1)^2 + b(1) + c \quad \Rightarrow \quad a + b + c = 2$

2. For $(2, 6)$:

$6 = a(2)^2 + b(2) + c \quad \Rightarrow \quad 4a + 2b + c = 6$

3. For $(-1, 18)$:

$18 = a(-1)^2 + b(-1) + c \quad \Rightarrow \quad a - b + c = 18$

We now have the following system of equations:

a + b + c &= 2 \quad \text{(1)} \\ 4a + 2b + c &= 6 \quad \text{(2)} \\ a - b + c &= 18 \quad \text{(3)} \end{align*}$$ Let's solve this system step by step. ### Step 1: Subtract Equation (1) from Equation (2) $$(4a + 2b + c) - (a + b + c) = 6 - 2$$ $$3a + b = 4 \quad \text{(4)}$$ ### Step 2: Subtract Equation (1) from Equation (3) $$(a - b + c) - (a + b + c) = 18 - 2$$ $$-2b = 16 \quad \Rightarrow \quad b = -8 \quad \text{(5)}$$ ### Step 3: Substitute $$b = -8$$ into Equation (4) $$3a + (-8) = 4$$ $$3a = 12 \quad \Rightarrow \quad a = 4 \quad \text{(6)}$$ ### Step 4: Substitute $$a = 4$$ and $$b = -8$$ into Equation (1) $$4 + (-8) + c = 2$$ $$-4 + c = 2 \quad \Rightarrow \quad c = 6 \quad \text{(7)}$$ ### Final Answer The values of $$a$$, $$b$$, and $$c$$ are: $$a = 4, \quad b = -8, \quad c = 6$$ Thus, the equation of the parabola is: $$y = 4x^2 - 8x + 6$$ Would you like more details or further explanation? Here are 5 related questions you might find useful: 1. How can you solve a system of equations using matrices? 2. What is the significance of the discriminant in quadratic equations? 3. How can you find the vertex of a parabola from its equation? 4. How does changing the coefficient $$a$$ affect the shape of a parabola? 5. What are the roots of this parabola, and how can we find them? **Tip:** When solving systems of equations, always check for consistent and independent solutions before proceeding to simplify!