Find the Equation of the Parabola (-2,8) , (0,-4) , (4,68)

$(- 2, 8)$ , $(0, - 4)$ , $(4, 68)$

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.

$8 = a {(- 2)}^{2} + b (- 2) + c, - 4 = a {(0)}^{2} + b (0) + c, 68 = a {(4)}^{2} + b (4) + c$

Step 3

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

Solve for $c$ in $- 4 = a \cdot 0^{2} + c$ .

Rewrite the equation as $a \cdot 0^{2} + c = - 4$ .

$a \cdot 0^{2} + c = - 4$

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

$68 = a \cdot 4^{2} + b (4) + c$

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

Simplify each term.

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

$a \cdot 0 + c = - 4$

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

$68 = a \cdot 4^{2} + b (4) + c$

Multiply $a$ by $0$ .

$0 + c = - 4$

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

$68 = a \cdot 4^{2} + b (4) + c$

$0 + c = - 4$

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

$68 = a \cdot 4^{2} + b (4) + c$

Add $0$ and $c$ .

$c = - 4$

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

$68 = a \cdot 4^{2} + b (4) + c$

$c = - 4$

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

$68 = a \cdot 4^{2} + b (4) + c$

$c = - 4$

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

$68 = a \cdot 4^{2} + b (4) + c$

Replace all occurrences of $c$ with $- 4$ in each equation.

Replace all occurrences of $c$ in $8 = a {(- 2)}^{2} + b (- 2) + c$ with $- 4$ .

$8 = a {(- 2)}^{2} + b (- 2) - 4$

$c = - 4$

$68 = a \cdot 4^{2} + b (4) + c$

Simplify $8 = a {(- 2)}^{2} + b (- 2) - 4$ .

Simplify the left side.

Remove parentheses.

$8 = a {(- 2)}^{2} + b (- 2) - 4$

$c = - 4$

$68 = a \cdot 4^{2} + b (4) + c$

$8 = a {(- 2)}^{2} + b (- 2) - 4$

$c = - 4$

$68 = a \cdot 4^{2} + b (4) + c$

Simplify the right side.

Simplify each term.

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

$8 = a \cdot 4 + b (- 2) - 4$

$c = - 4$

$68 = a \cdot 4^{2} + b (4) + c$

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

$8 = 4 \cdot a + b (- 2) - 4$

$c = - 4$

$68 = a \cdot 4^{2} + b (4) + c$

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

$8 = 4 a - 2 b - 4$

$c = - 4$

$68 = a \cdot 4^{2} + b (4) + c$

$8 = 4 a - 2 b - 4$

$c = - 4$

$68 = a \cdot 4^{2} + b (4) + c$

$8 = 4 a - 2 b - 4$

$c = - 4$

$68 = a \cdot 4^{2} + b (4) + c$

$8 = 4 a - 2 b - 4$

$c = - 4$

$68 = a \cdot 4^{2} + b (4) + c$

Replace all occurrences of $c$ in $68 = a \cdot 4^{2} + b (4) + c$ with $- 4$ .

$68 = a \cdot 4^{2} + b (4) - 4$

$8 = 4 a - 2 b - 4$

$c = - 4$

Simplify $68 = a \cdot 4^{2} + b (4) - 4$ .

Simplify the left side.

Remove parentheses.

$68 = a \cdot 4^{2} + b (4) - 4$

$8 = 4 a - 2 b - 4$

$c = - 4$

$68 = a \cdot 4^{2} + b (4) - 4$

$8 = 4 a - 2 b - 4$

$c = - 4$

Simplify the right side.

Simplify each term.

Raise $4$ to the power of $2$ .

$68 = a \cdot 16 + b (4) - 4$

$8 = 4 a - 2 b - 4$

$c = - 4$

Move $16$ to the left of $a$ .

$68 = 16 \cdot a + b (4) - 4$

$8 = 4 a - 2 b - 4$

$c = - 4$

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

$68 = 16 a + 4 b - 4$

$8 = 4 a - 2 b - 4$

$c = - 4$

$68 = 16 a + 4 b - 4$

$8 = 4 a - 2 b - 4$

$c = - 4$

$68 = 16 a + 4 b - 4$

$8 = 4 a - 2 b - 4$

$c = - 4$

$68 = 16 a + 4 b - 4$

$8 = 4 a - 2 b - 4$

$c = - 4$

$68 = 16 a + 4 b - 4$

$8 = 4 a - 2 b - 4$

$c = - 4$

Solve for $a$ in $68 = 16 a + 4 b - 4$ .

Rewrite the equation as $16 a + 4 b - 4 = 68$ .

$16 a + 4 b - 4 = 68$

$8 = 4 a - 2 b - 4$

$c = - 4$

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

Subtract $4 b$ from both sides of the equation.

$16 a - 4 = 68 - 4 b$

$8 = 4 a - 2 b - 4$

$c = - 4$

Add $4$ to both sides of the equation.

$16 a = 68 - 4 b + 4$

$8 = 4 a - 2 b - 4$

$c = - 4$

Add $68$ and $4$ .

$16 a = - 4 b + 72$

$8 = 4 a - 2 b - 4$

$c = - 4$

$16 a = - 4 b + 72$

$8 = 4 a - 2 b - 4$

$c = - 4$

Divide each term in $16 a = - 4 b + 72$ by $16$ and simplify.

Divide each term in $16 a = - 4 b + 72$ by $16$ .

$\frac{16 a}{16} = \frac{- 4 b}{16} + \frac{72}{16}$

$8 = 4 a - 2 b - 4$

$c = - 4$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 a}{16} = \frac{- 4 b}{16} + \frac{72}{16}$

$8 = 4 a - 2 b - 4$

$c = - 4$

Divide $a$ by $1$ .

$a = \frac{- 4 b}{16} + \frac{72}{16}$

$8 = 4 a - 2 b - 4$

$c = - 4$

$a = \frac{- 4 b}{16} + \frac{72}{16}$

$8 = 4 a - 2 b - 4$

$c = - 4$

$a = \frac{- 4 b}{16} + \frac{72}{16}$

$8 = 4 a - 2 b - 4$

$c = - 4$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 4$ and $16$ .

Factor $4$ out of $- 4 b$ .

$a = \frac{4 (- b)}{16} + \frac{72}{16}$

$8 = 4 a - 2 b - 4$

$c = - 4$

Cancel the common factors.

Factor $4$ out of $16$ .

$a = \frac{4 (- b)}{4 (4)} + \frac{72}{16}$

$8 = 4 a - 2 b - 4$

$c = - 4$

Cancel the common factor.

$a = \frac{4 (- b)}{4 \cdot 4} + \frac{72}{16}$

$8 = 4 a - 2 b - 4$

$c = - 4$

Rewrite the expression.

$a = \frac{- b}{4} + \frac{72}{16}$

$8 = 4 a - 2 b - 4$

$c = - 4$

$a = \frac{- b}{4} + \frac{72}{16}$

$8 = 4 a - 2 b - 4$

$c = - 4$

$a = \frac{- b}{4} + \frac{72}{16}$

$8 = 4 a - 2 b - 4$

$c = - 4$

Move the negative in front of the fraction.

$a = - \frac{b}{4} + \frac{72}{16}$

$8 = 4 a - 2 b - 4$

$c = - 4$

Cancel the common factor of $72$ and $16$ .

Factor $8$ out of $72$ .

$a = - \frac{b}{4} + \frac{8 (9)}{16}$

$8 = 4 a - 2 b - 4$

$c = - 4$

Cancel the common factors.

Factor $8$ out of $16$ .

$a = - \frac{b}{4} + \frac{8 \cdot 9}{8 \cdot 2}$

$8 = 4 a - 2 b - 4$

$c = - 4$

Cancel the common factor.

$a = - \frac{b}{4} + \frac{8 \cdot 9}{8 \cdot 2}$

$8 = 4 a - 2 b - 4$

$c = - 4$

Rewrite the expression.

$a = - \frac{b}{4} + \frac{9}{2}$

$8 = 4 a - 2 b - 4$

$c = - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$8 = 4 a - 2 b - 4$

$c = - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$8 = 4 a - 2 b - 4$

$c = - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$8 = 4 a - 2 b - 4$

$c = - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$8 = 4 a - 2 b - 4$

$c = - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$8 = 4 a - 2 b - 4$

$c = - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$8 = 4 a - 2 b - 4$

$c = - 4$

Replace all occurrences of $a$ with $- \frac{b}{4} + \frac{9}{2}$ in each equation.

Replace all occurrences of $a$ in $8 = 4 a - 2 b - 4$ with $- \frac{b}{4} + \frac{9}{2}$ .

$8 = 4 (- \frac{b}{4} + \frac{9}{2}) - 2 b - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Simplify the right side.

Simplify $4 (- \frac{b}{4} + \frac{9}{2}) - 2 b - 4$ .

Simplify each term.

Apply the distributive property.

$8 = 4 (- \frac{b}{4}) + 4 (\frac{9}{2}) - 2 b - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{b}{4}$ into the numerator.

$8 = 4 (\frac{- b}{4}) + 4 (\frac{9}{2}) - 2 b - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Cancel the common factor.

$8 = 4 (\frac{- b}{4}) + 4 (\frac{9}{2}) - 2 b - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Rewrite the expression.

$8 = - b + 4 (\frac{9}{2}) - 2 b - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

$8 = - b + 4 (\frac{9}{2}) - 2 b - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$8 = - b + 2 (2) (\frac{9}{2}) - 2 b - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Cancel the common factor.

$8 = - b + 2 \cdot (2 (\frac{9}{2})) - 2 b - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Rewrite the expression.

$8 = - b + 2 \cdot 9 - 2 b - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

$8 = - b + 2 \cdot 9 - 2 b - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Multiply $2$ by $9$ .

$8 = - b + 18 - 2 b - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

$8 = - b + 18 - 2 b - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Simplify by adding terms.

Subtract $2 b$ from $- b$ .

$8 = - 3 b + 18 - 4$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Subtract $4$ from $18$ .

$8 = - 3 b + 14$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

$8 = - 3 b + 14$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

$8 = - 3 b + 14$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

$8 = - 3 b + 14$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

$8 = - 3 b + 14$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Solve for $b$ in $8 = - 3 b + 14$ .

Rewrite the equation as $- 3 b + 14 = 8$ .

$- 3 b + 14 = 8$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

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

Subtract $14$ from both sides of the equation.

$- 3 b = 8 - 14$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Subtract $14$ from $8$ .

$- 3 b = - 6$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

$- 3 b = - 6$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Divide each term in $- 3 b = - 6$ by $- 3$ and simplify.

Divide each term in $- 3 b = - 6$ by $- 3$ .

$\frac{- 3 b}{- 3} = \frac{- 6}{- 3}$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 b}{- 3} = \frac{- 6}{- 3}$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Divide $b$ by $1$ .

$b = \frac{- 6}{- 3}$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

$b = \frac{- 6}{- 3}$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

$b = \frac{- 6}{- 3}$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

Simplify the right side.

Divide $- 6$ by $- 3$ .

$b = 2$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

$b = 2$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

$b = 2$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

$b = 2$

$a = - \frac{b}{4} + \frac{9}{2}$

$c = - 4$

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

Replace all occurrences of $b$ in $a = - \frac{b}{4} + \frac{9}{2}$ with $2$ .

$a = - \frac{2}{4} + \frac{9}{2}$

$b = 2$

$c = - 4$

Simplify the right side.

Simplify $- \frac{2}{4} + \frac{9}{2}$ .

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$a = - \frac{2 (1)}{4} + \frac{9}{2}$

$b = 2$

$c = - 4$

Cancel the common factors.

Factor $2$ out of $4$ .

$a = - \frac{2 \cdot 1}{2 \cdot 2} + \frac{9}{2}$

$b = 2$

$c = - 4$

Cancel the common factor.

$a = - \frac{2 \cdot 1}{2 \cdot 2} + \frac{9}{2}$

$b = 2$

$c = - 4$

Rewrite the expression.

$a = - \frac{1}{2} + \frac{9}{2}$

$b = 2$

$c = - 4$

$a = - \frac{1}{2} + \frac{9}{2}$

$b = 2$

$c = - 4$

$a = - \frac{1}{2} + \frac{9}{2}$

$b = 2$

$c = - 4$

Combine the numerators over the common denominator.

$a = \frac{- 1 + 9}{2}$

$b = 2$

$c = - 4$

Simplify the expression.

Add $- 1$ and $9$ .

$a = \frac{8}{2}$

$b = 2$

$c = - 4$

Divide $8$ by $2$ .

$a = 4$

$b = 2$

$c = - 4$

$a = 4$

$b = 2$

$c = - 4$

$a = 4$

$b = 2$

$c = - 4$

$a = 4$

$b = 2$

$c = - 4$

$a = 4$

$b = 2$

$c = - 4$

List all of the solutions.

$a = 4, b = 2, c = - 4$

Step 4

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

$y = 4 x^{2} + 2 x - 4$

Step 5

Find the Equation of the Parabola (-2,8) , (0,-4) , (4,68)

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