Find the Equation of the Parabola (0,5) , (3,8) , (-1,8)

$(0, 5)$ , $(3, 8)$ , $(- 1, 8)$

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.

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

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.

$5 = a {(0)}^{2} + b (0) + c$

$8 = a {(3)}^{2} + b (3) + c$

$8 = a {(- 1)}^{2} + b (- 1) + c$

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

Solve for $c$ in $5 = a \cdot 0^{2} + c$ .

Rewrite the equation as $a \cdot 0^{2} + c = 5$ .

$a \cdot 0^{2} + c = 5$

$8 = a \cdot 3^{2} + b (3) + c$

$8 = a {(- 1)}^{2} + b (- 1) + c$

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

Simplify each term.

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

$a \cdot 0 + c = 5$

$8 = a \cdot 3^{2} + b (3) + c$

$8 = a {(- 1)}^{2} + b (- 1) + c$

Multiply $a$ by $0$ .

$0 + c = 5$

$8 = a \cdot 3^{2} + b (3) + c$

$8 = a {(- 1)}^{2} + b (- 1) + c$

$0 + c = 5$

$8 = a \cdot 3^{2} + b (3) + c$

$8 = a {(- 1)}^{2} + b (- 1) + c$

Add $0$ and $c$ .

$c = 5$

$8 = a \cdot 3^{2} + b (3) + c$

$8 = a {(- 1)}^{2} + b (- 1) + c$

$c = 5$

$8 = a \cdot 3^{2} + b (3) + c$

$8 = a {(- 1)}^{2} + b (- 1) + c$

$c = 5$

$8 = a \cdot 3^{2} + b (3) + c$

$8 = a {(- 1)}^{2} + b (- 1) + c$

Replace all occurrences of $c$ with $5$ in each equation.

Replace all occurrences of $c$ in $8 = a \cdot 3^{2} + b (3) + c$ with $5$ .

$8 = a \cdot 3^{2} + b (3) + 5$

$c = 5$

$8 = a {(- 1)}^{2} + b (- 1) + c$

Simplify $8 = a \cdot 3^{2} + b (3) + 5$ .

Simplify the left side.

Remove parentheses.

$8 = a \cdot 3^{2} + b (3) + 5$

$c = 5$

$8 = a {(- 1)}^{2} + b (- 1) + c$

$8 = a \cdot 3^{2} + b (3) + 5$

$c = 5$

$8 = a {(- 1)}^{2} + b (- 1) + c$

Simplify the right side.

Simplify each term.

Raise $3$ to the power of $2$ .

$8 = a \cdot 9 + b (3) + 5$

$c = 5$

$8 = a {(- 1)}^{2} + b (- 1) + c$

Move $9$ to the left of $a$ .

$8 = 9 \cdot a + b (3) + 5$

$c = 5$

$8 = a {(- 1)}^{2} + b (- 1) + c$

Move $3$ to the left of $b$ .

$8 = 9 a + 3 b + 5$

$c = 5$

$8 = a {(- 1)}^{2} + b (- 1) + c$

$8 = 9 a + 3 b + 5$

$c = 5$

$8 = a {(- 1)}^{2} + b (- 1) + c$

$8 = 9 a + 3 b + 5$

$c = 5$

$8 = a {(- 1)}^{2} + b (- 1) + c$

$8 = 9 a + 3 b + 5$

$c = 5$

$8 = a {(- 1)}^{2} + b (- 1) + c$

Replace all occurrences of $c$ in $8 = a {(- 1)}^{2} + b (- 1) + c$ with $5$ .

$8 = a {(- 1)}^{2} + b (- 1) + 5$

$8 = 9 a + 3 b + 5$

$c = 5$

Simplify $8 = a {(- 1)}^{2} + b (- 1) + 5$ .

Simplify the left side.

Remove parentheses.

$8 = a {(- 1)}^{2} + b (- 1) + 5$

$8 = 9 a + 3 b + 5$

$c = 5$

$8 = a {(- 1)}^{2} + b (- 1) + 5$

$8 = 9 a + 3 b + 5$

$c = 5$

Simplify the right side.

Simplify each term.

Raise $- 1$ to the power of $2$ .

$8 = a \cdot 1 + b (- 1) + 5$

$8 = 9 a + 3 b + 5$

$c = 5$

Multiply $a$ by $1$ .

$8 = a + b (- 1) + 5$

$8 = 9 a + 3 b + 5$

$c = 5$

Move $- 1$ to the left of $b$ .

$8 = a - 1 b + 5$

$8 = 9 a + 3 b + 5$

$c = 5$

Rewrite $- 1 b$ as $- b$ .

$8 = a - b + 5$

$8 = 9 a + 3 b + 5$

$c = 5$

$8 = a - b + 5$

$8 = 9 a + 3 b + 5$

$c = 5$

$8 = a - b + 5$

$8 = 9 a + 3 b + 5$

$c = 5$

$8 = a - b + 5$

$8 = 9 a + 3 b + 5$

$c = 5$

$8 = a - b + 5$

$8 = 9 a + 3 b + 5$

$c = 5$

Solve for $a$ in $8 = a - b + 5$ .

Rewrite the equation as $a - b + 5 = 8$ .

$a - b + 5 = 8$

$8 = 9 a + 3 b + 5$

$c = 5$

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

Add $b$ to both sides of the equation.

$a + 5 = 8 + b$

$8 = 9 a + 3 b + 5$

$c = 5$

Subtract $5$ from both sides of the equation.

$a = 8 + b - 5$

$8 = 9 a + 3 b + 5$

$c = 5$

Subtract $5$ from $8$ .

$a = b + 3$

$8 = 9 a + 3 b + 5$

$c = 5$

$a = b + 3$

$8 = 9 a + 3 b + 5$

$c = 5$

$a = b + 3$

$8 = 9 a + 3 b + 5$

$c = 5$

Replace all occurrences of $a$ with $b + 3$ in each equation.

Replace all occurrences of $a$ in $8 = 9 a + 3 b + 5$ with $b + 3$ .

$8 = 9 (b + 3) + 3 b + 5$

$a = b + 3$

$c = 5$

Simplify the right side.

Simplify $9 (b + 3) + 3 b + 5$ .

Simplify each term.

Apply the distributive property.

$8 = 9 b + 9 \cdot 3 + 3 b + 5$

$a = b + 3$

$c = 5$

Multiply $9$ by $3$ .

$8 = 9 b + 27 + 3 b + 5$

$a = b + 3$

$c = 5$

$8 = 9 b + 27 + 3 b + 5$

$a = b + 3$

$c = 5$

Simplify by adding terms.

Add $9 b$ and $3 b$ .

$8 = 12 b + 27 + 5$

$a = b + 3$

$c = 5$

Add $27$ and $5$ .

$8 = 12 b + 32$

$a = b + 3$

$c = 5$

$8 = 12 b + 32$

$a = b + 3$

$c = 5$

$8 = 12 b + 32$

$a = b + 3$

$c = 5$

$8 = 12 b + 32$

$a = b + 3$

$c = 5$

$8 = 12 b + 32$

$a = b + 3$

$c = 5$

Solve for $b$ in $8 = 12 b + 32$ .

Rewrite the equation as $12 b + 32 = 8$ .

$12 b + 32 = 8$

$a = b + 3$

$c = 5$

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

Subtract $32$ from both sides of the equation.

$12 b = 8 - 32$

$a = b + 3$

$c = 5$

Subtract $32$ from $8$ .

$12 b = - 24$

$a = b + 3$

$c = 5$

$12 b = - 24$

$a = b + 3$

$c = 5$

Divide each term in $12 b = - 24$ by $12$ and simplify.

Divide each term in $12 b = - 24$ by $12$ .

$\frac{12 b}{12} = \frac{- 24}{12}$

$a = b + 3$

$c = 5$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 b}{12} = \frac{- 24}{12}$

$a = b + 3$

$c = 5$

Divide $b$ by $1$ .

$b = \frac{- 24}{12}$

$a = b + 3$

$c = 5$

$b = \frac{- 24}{12}$

$a = b + 3$

$c = 5$

$b = \frac{- 24}{12}$

$a = b + 3$

$c = 5$

Simplify the right side.

Divide $- 24$ by $12$ .

$b = - 2$

$a = b + 3$

$c = 5$

$b = - 2$

$a = b + 3$

$c = 5$

$b = - 2$

$a = b + 3$

$c = 5$

$b = - 2$

$a = b + 3$

$c = 5$

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

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

$a = (- 2) + 3$

$b = - 2$

$c = 5$

Simplify the right side.

Add $- 2$ and $3$ .

$a = 1$

$b = - 2$

$c = 5$

$a = 1$

$b = - 2$

$c = 5$

$a = 1$

$b = - 2$

$c = 5$

List all of the solutions.

$a = 1, b = - 2, c = 5$

$a = 1, b = - 2, c = 5$

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

$y = x^{2} - 2 x + 5$

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