Find the Equation of the Parabola (0,3) , (1,2) , (2,3)

$(0, 3)$ , $(1, 2)$ , $(2, 3)$

Step 1

Use the standard form of a quadratic equation $y = a x^{2} + b x + c$ as the starting point for finding the equation through the three points.

$y = a x^{2} + b x + c$

Step 2

Create a system of equations by substituting the $x$ and $y$ values of each point into the standard formula of a quadratic equation to create the three equation system.

$3 = a {(0)}^{2} + b (0) + c$

$2 = a {(1)}^{2} + b (1) + c$

$3 = a {(2)}^{2} + b (2) + c$

Step 3

Solve the system of equations to find the values of $a$ , $b$ , and $c$ .

Solve for $c$ in $3 = a \cdot 0^{2} + c$ .

Rewrite the equation as $a \cdot 0^{2} + c = 3$ .

$a \cdot 0^{2} + c = 3$

$2 = a + b + c$

$3 = a \cdot 2^{2} + b (2) + c$

Simplify $a \cdot 0^{2} + c$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$a \cdot 0 + c = 3$

$2 = a + b + c$

$3 = a \cdot 2^{2} + b (2) + c$

Multiply $a$ by $0$ .

$0 + c = 3$

$2 = a + b + c$

$3 = a \cdot 2^{2} + b (2) + c$

$0 + c = 3$

$2 = a + b + c$

$3 = a \cdot 2^{2} + b (2) + c$

Add $0$ and $c$ .

$c = 3$

$2 = a + b + c$

$3 = a \cdot 2^{2} + b (2) + c$

$c = 3$

$2 = a + b + c$

$3 = a \cdot 2^{2} + b (2) + c$

$c = 3$

$2 = a + b + c$

$3 = a \cdot 2^{2} + b (2) + c$

Replace all occurrences of $c$ with $3$ in each equation.

Replace all occurrences of $c$ in $2 = a + b + c$ with $3$ .

$2 = a + b + 3$

$c = 3$

$3 = a \cdot 2^{2} + b (2) + c$

Simplify the left side.

Remove parentheses.

$2 = a + b + 3$

$c = 3$

$3 = a \cdot 2^{2} + b (2) + c$

$2 = a + b + 3$

$c = 3$

$3 = a \cdot 2^{2} + b (2) + c$

Replace all occurrences of $c$ in $3 = a \cdot 2^{2} + b (2) + c$ with $3$ .

$3 = a \cdot 2^{2} + b (2) + 3$

$2 = a + b + 3$

$c = 3$

Simplify $3 = a \cdot 2^{2} + b (2) + 3$ .

Simplify the left side.

Remove parentheses.

$3 = a \cdot 2^{2} + b (2) + 3$

$2 = a + b + 3$

$c = 3$

$3 = a \cdot 2^{2} + b (2) + 3$

$2 = a + b + 3$

$c = 3$

Simplify the right side.

Simplify each term.

Raise $2$ to the power of $2$ .

$3 = a \cdot 4 + b (2) + 3$

$2 = a + b + 3$

$c = 3$

Move $4$ to the left of $a$ .

$3 = 4 \cdot a + b (2) + 3$

$2 = a + b + 3$

$c = 3$

Move $2$ to the left of $b$ .

$3 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

$3 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

$3 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

$3 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

$3 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

Solve for $a$ in $2 = a + b + 3$ .

Rewrite the equation as $a + b + 3 = 2$ .

$a + b + 3 = 2$

$3 = 4 a + 2 b + 3$

$c = 3$

Move all terms not containing $a$ to the right side of the equation.

Subtract $b$ from both sides of the equation.

$a + 3 = 2 - b$

$3 = 4 a + 2 b + 3$

$c = 3$

Subtract $3$ from both sides of the equation.

$a = 2 - b - 3$

$3 = 4 a + 2 b + 3$

$c = 3$

Subtract $3$ from $2$ .

$a = - b - 1$

$3 = 4 a + 2 b + 3$

$c = 3$

$a = - b - 1$

$3 = 4 a + 2 b + 3$

$c = 3$

$a = - b - 1$

$3 = 4 a + 2 b + 3$

$c = 3$

Replace all occurrences of $a$ with $- b - 1$ in each equation.

Replace all occurrences of $a$ in $3 = 4 a + 2 b + 3$ with $- b - 1$ .

$3 = 4 (- b - 1) + 2 b + 3$

$a = - b - 1$

$c = 3$

Simplify the right side.

Simplify $4 (- b - 1) + 2 b + 3$ .

Simplify each term.

Apply the distributive property.

$3 = 4 (- b) + 4 \cdot - 1 + 2 b + 3$

$a = - b - 1$

$c = 3$

Multiply $- 1$ by $4$ .

$3 = - 4 b + 4 \cdot - 1 + 2 b + 3$

$a = - b - 1$

$c = 3$

Multiply $4$ by $- 1$ .

$3 = - 4 b - 4 + 2 b + 3$

$a = - b - 1$

$c = 3$

$3 = - 4 b - 4 + 2 b + 3$

$a = - b - 1$

$c = 3$

Simplify by adding terms.

Add $- 4 b$ and $2 b$ .

$3 = - 2 b - 4 + 3$

$a = - b - 1$

$c = 3$

Add $- 4$ and $3$ .

$3 = - 2 b - 1$

$a = - b - 1$

$c = 3$

$3 = - 2 b - 1$

$a = - b - 1$

$c = 3$

$3 = - 2 b - 1$

$a = - b - 1$

$c = 3$

$3 = - 2 b - 1$

$a = - b - 1$

$c = 3$

$3 = - 2 b - 1$

$a = - b - 1$

$c = 3$

Solve for $b$ in $3 = - 2 b - 1$ .

Rewrite the equation as $- 2 b - 1 = 3$ .

$- 2 b - 1 = 3$

$a = - b - 1$

$c = 3$

Move all terms not containing $b$ to the right side of the equation.

Add $1$ to both sides of the equation.

$- 2 b = 3 + 1$

$a = - b - 1$

$c = 3$

Add $3$ and $1$ .

$- 2 b = 4$

$a = - b - 1$

$c = 3$

$- 2 b = 4$

$a = - b - 1$

$c = 3$

Divide each term in $- 2 b = 4$ by $- 2$ and simplify.

Divide each term in $- 2 b = 4$ by $- 2$ .

$\frac{- 2 b}{- 2} = \frac{4}{- 2}$

$a = - b - 1$

$c = 3$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 b}{- 2} = \frac{4}{- 2}$

$a = - b - 1$

$c = 3$

Divide $b$ by $1$ .

$b = \frac{4}{- 2}$

$a = - b - 1$

$c = 3$

$b = \frac{4}{- 2}$

$a = - b - 1$

$c = 3$

$b = \frac{4}{- 2}$

$a = - b - 1$

$c = 3$

Simplify the right side.

Divide $4$ by $- 2$ .

$b = - 2$

$a = - b - 1$

$c = 3$

$b = - 2$

$a = - b - 1$

$c = 3$

$b = - 2$

$a = - b - 1$

$c = 3$

$b = - 2$

$a = - b - 1$

$c = 3$

Replace all occurrences of $b$ with $- 2$ in each equation.

Replace all occurrences of $b$ in $a = - b - 1$ with $- 2$ .

$a = - (- 2) - 1$

$b = - 2$

$c = 3$

Simplify the right side.

Simplify $- (- 2) - 1$ .

Multiply $- 1$ by $- 2$ .

$a = 2 - 1$

$b = - 2$

$c = 3$

Subtract $1$ from $2$ .

$a = 1$

$b = - 2$

$c = 3$

$a = 1$

$b = - 2$

$c = 3$

$a = 1$

$b = - 2$

$c = 3$

$a = 1$

$b = - 2$

$c = 3$

List all of the solutions.

$a = 1, b = - 2, c = 3$

$a = 1, b = - 2, c = 3$

Step 4

Substitute the actual values of $a$ , $b$ , and $c$ into the formula for a quadratic equation to find the resulting equation.

$y = x^{2} - 2 x + 3$

Step 5

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