Find the Function Rule table[[x,y],[1,66],[2,72],[3,82]]

$\begin{array}{c} x & y \\ 1 & 66 \\ 2 & 72 \\ 3 & 82 \end{array}$

Check if the function rule is linear.

To find if the table follows a function rule, check to see if the values follow the linear form $y = a x + b$ .

$y = a x + b$

Build a set of equations from the table such that $y = a x + b$ .

$66 = a (1) + b 72 = a (2) + b 82 = a (3) + b$

Calculate the values of $a$ and $b$ .

Solve for $a$ in $66 = a + b$ .

Rewrite the equation as $a + b = 66$ .

$a + b = 66$

$72 = a (2) + b$

$82 = a (3) + b$

Subtract $b$ from both sides of the equation.

$a = 66 - b$

$72 = a (2) + b$

$82 = a (3) + b$

$a = 66 - b$

$72 = a (2) + b$

$82 = a (3) + b$

Replace all occurrences of $a$ with $66 - b$ in each equation.

Replace all occurrences of $a$ in $72 = a (2) + b$ with $66 - b$ .

$72 = (66 - b) (2) + b$

$a = 66 - b$

$82 = a (3) + b$

Simplify the right side.

Simplify $(66 - b) (2) + b$ .

Simplify each term.

Apply the distributive property.

$72 = 66 \cdot 2 - b \cdot 2 + b$

$a = 66 - b$

$82 = a (3) + b$

Multiply $66$ by $2$ .

$72 = 132 - b \cdot 2 + b$

$a = 66 - b$

$82 = a (3) + b$

Multiply $2$ by $- 1$ .

$72 = 132 - 2 b + b$

$a = 66 - b$

$82 = a (3) + b$

$72 = 132 - 2 b + b$

$a = 66 - b$

$82 = a (3) + b$

Add $- 2 b$ and $b$ .

$72 = 132 - b$

$a = 66 - b$

$82 = a (3) + b$

$72 = 132 - b$

$a = 66 - b$

$82 = a (3) + b$

$72 = 132 - b$

$a = 66 - b$

$82 = a (3) + b$

Replace all occurrences of $a$ in $82 = a (3) + b$ with $66 - b$ .

$82 = (66 - b) (3) + b$

$72 = 132 - b$

$a = 66 - b$

Simplify the right side.

Simplify $(66 - b) (3) + b$ .

Simplify each term.

Apply the distributive property.

$82 = 66 \cdot 3 - b \cdot 3 + b$

$72 = 132 - b$

$a = 66 - b$

Multiply $66$ by $3$ .

$82 = 198 - b \cdot 3 + b$

$72 = 132 - b$

$a = 66 - b$

Multiply $3$ by $- 1$ .

$82 = 198 - 3 b + b$

$72 = 132 - b$

$a = 66 - b$

$82 = 198 - 3 b + b$

$72 = 132 - b$

$a = 66 - b$

Add $- 3 b$ and $b$ .

$82 = 198 - 2 b$

$72 = 132 - b$

$a = 66 - b$

$82 = 198 - 2 b$

$72 = 132 - b$

$a = 66 - b$

$82 = 198 - 2 b$

$72 = 132 - b$

$a = 66 - b$

$82 = 198 - 2 b$

$72 = 132 - b$

$a = 66 - b$

Solve for $b$ in $82 = 198 - 2 b$ .

Rewrite the equation as $198 - 2 b = 82$ .

$198 - 2 b = 82$

$72 = 132 - b$

$a = 66 - b$

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

Subtract $198$ from both sides of the equation.

$- 2 b = 82 - 198$

$72 = 132 - b$

$a = 66 - b$

Subtract $198$ from $82$ .

$- 2 b = - 116$

$72 = 132 - b$

$a = 66 - b$

$- 2 b = - 116$

$72 = 132 - b$

$a = 66 - b$

Divide each term in $- 2 b = - 116$ by $- 2$ and simplify.

Divide each term in $- 2 b = - 116$ by $- 2$ .

$\frac{- 2 b}{- 2} = \frac{- 116}{- 2}$

$72 = 132 - b$

$a = 66 - b$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 b}{- 2} = \frac{- 116}{- 2}$

$72 = 132 - b$

$a = 66 - b$

Divide $b$ by $1$ .

$b = \frac{- 116}{- 2}$

$72 = 132 - b$

$a = 66 - b$

$b = \frac{- 116}{- 2}$

$72 = 132 - b$

$a = 66 - b$

$b = \frac{- 116}{- 2}$

$72 = 132 - b$

$a = 66 - b$

Simplify the right side.

Divide $- 116$ by $- 2$ .

$b = 58$

$72 = 132 - b$

$a = 66 - b$

$b = 58$

$72 = 132 - b$

$a = 66 - b$

$b = 58$

$72 = 132 - b$

$a = 66 - b$

$b = 58$

$72 = 132 - b$

$a = 66 - b$

Replace all occurrences of $b$ with $58$ in each equation.

Replace all occurrences of $b$ in $72 = 132 - b$ with $58$ .

$72 = 132 - (58)$

$b = 58$

$a = 66 - b$

Simplify the right side.

Simplify $132 - (58)$ .

Multiply $- 1$ by $58$ .

$72 = 132 - 58$

$b = 58$

$a = 66 - b$

Subtract $58$ from $132$ .

$72 = 74$

$b = 58$

$a = 66 - b$

$72 = 74$

$b = 58$

$a = 66 - b$

$72 = 74$

$b = 58$

$a = 66 - b$

Replace all occurrences of $b$ in $a = 66 - b$ with $58$ .

$a = 66 - (58)$

$72 = 74$

$b = 58$

Simplify the right side.

Simplify $66 - (58)$ .

Multiply $- 1$ by $58$ .

$a = 66 - 58$

$72 = 74$

$b = 58$

Subtract $58$ from $66$ .

$a = 8$

$72 = 74$

$b = 58$

$a = 8$

$72 = 74$

$b = 58$

$a = 8$

$72 = 74$

$b = 58$

$a = 8$

$72 = 74$

$b = 58$

Since $72 = 74$ is not true, there is no solution.

No solution

Since $y \neq y$ for the corresponding $x$ values, the function is not linear.

The function is not linear

Check if the function rule is quadratic.

To find if the table follows a function rule, check whether the function rule could follow the form $y = a x^{2} + b x + c$ .

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

Build a set of $3$ equations from the table such that $y = a x^{2} + b x + c$ .

Calculate the values of $a$ , $b$ , and $c$ .

Solve for $a$ in $66 = a + b + c$ .

Rewrite the equation as $a + b + c = 66$ .

$a + b + c = 66$

$72 = a \cdot 2^{2} + b (2) + c$

$82 = a \cdot 3^{2} + b (3) + c$

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

Subtract $b$ from both sides of the equation.

$a + c = 66 - b$

$72 = a \cdot 2^{2} + b (2) + c$

$82 = a \cdot 3^{2} + b (3) + c$

Subtract $c$ from both sides of the equation.

$a = 66 - b - c$

$72 = a \cdot 2^{2} + b (2) + c$

$82 = a \cdot 3^{2} + b (3) + c$

$a = 66 - b - c$

$72 = a \cdot 2^{2} + b (2) + c$

$82 = a \cdot 3^{2} + b (3) + c$

$a = 66 - b - c$

$72 = a \cdot 2^{2} + b (2) + c$

$82 = a \cdot 3^{2} + b (3) + c$

Replace all occurrences of $a$ with $66 - b - c$ in each equation.

Replace all occurrences of $a$ in $72 = a \cdot 2^{2} + b (2) + c$ with $66 - b - c$ .

$72 = (66 - b - c) \cdot 2^{2} + b (2) + c$

$a = 66 - b - c$

$82 = a \cdot 3^{2} + b (3) + c$

Simplify the right side.

Simplify $(66 - b - c) \cdot 2^{2} + b (2) + c$ .

Simplify each term.

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

$72 = (66 - b - c) \cdot 4 + b (2) + c$

$a = 66 - b - c$

$82 = a \cdot 3^{2} + b (3) + c$

Apply the distributive property.

$72 = 66 \cdot 4 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = 66 - b - c$

$82 = a \cdot 3^{2} + b (3) + c$

Simplify.

Multiply $66$ by $4$ .

$72 = 264 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = 66 - b - c$

$82 = a \cdot 3^{2} + b (3) + c$

Multiply $4$ by $- 1$ .

$72 = 264 - 4 b - c \cdot 4 + b (2) + c$

$a = 66 - b - c$

$82 = a \cdot 3^{2} + b (3) + c$

Multiply $4$ by $- 1$ .

$72 = 264 - 4 b - 4 c + b (2) + c$

$a = 66 - b - c$

$82 = a \cdot 3^{2} + b (3) + c$

$72 = 264 - 4 b - 4 c + b (2) + c$

$a = 66 - b - c$

$82 = a \cdot 3^{2} + b (3) + c$

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

$72 = 264 - 4 b - 4 c + 2 b + c$

$a = 66 - b - c$

$82 = a \cdot 3^{2} + b (3) + c$

$72 = 264 - 4 b - 4 c + 2 b + c$

$a = 66 - b - c$

$82 = a \cdot 3^{2} + b (3) + c$

Simplify by adding terms.

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

$72 = 264 - 2 b - 4 c + c$

$a = 66 - b - c$

$82 = a \cdot 3^{2} + b (3) + c$

Add $- 4 c$ and $c$ .

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$82 = a \cdot 3^{2} + b (3) + c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$82 = a \cdot 3^{2} + b (3) + c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$82 = a \cdot 3^{2} + b (3) + c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$82 = a \cdot 3^{2} + b (3) + c$

Replace all occurrences of $a$ in $82 = a \cdot 3^{2} + b (3) + c$ with $66 - b - c$ .

$82 = (66 - b - c) \cdot 3^{2} + b (3) + c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Simplify the right side.

Simplify $(66 - b - c) \cdot 3^{2} + b (3) + c$ .

Simplify each term.

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

$82 = (66 - b - c) \cdot 9 + b (3) + c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Apply the distributive property.

$82 = 66 \cdot 9 - b \cdot 9 - c \cdot 9 + b (3) + c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Simplify.

Multiply $66$ by $9$ .

$82 = 594 - b \cdot 9 - c \cdot 9 + b (3) + c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Multiply $9$ by $- 1$ .

$82 = 594 - 9 b - c \cdot 9 + b (3) + c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Multiply $9$ by $- 1$ .

$82 = 594 - 9 b - 9 c + b (3) + c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$82 = 594 - 9 b - 9 c + b (3) + c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

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

$82 = 594 - 9 b - 9 c + 3 b + c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$82 = 594 - 9 b - 9 c + 3 b + c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Simplify by adding terms.

Add $- 9 b$ and $3 b$ .

$82 = 594 - 6 b - 9 c + c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Add $- 9 c$ and $c$ .

$82 = 594 - 6 b - 8 c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$82 = 594 - 6 b - 8 c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$82 = 594 - 6 b - 8 c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$82 = 594 - 6 b - 8 c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$82 = 594 - 6 b - 8 c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Solve for $b$ in $82 = 594 - 6 b - 8 c$ .

Rewrite the equation as $594 - 6 b - 8 c = 82$ .

$594 - 6 b - 8 c = 82$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

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

Subtract $594$ from both sides of the equation.

$- 6 b - 8 c = 82 - 594$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Add $8 c$ to both sides of the equation.

$- 6 b = 82 - 594 + 8 c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Subtract $594$ from $82$ .

$- 6 b = - 512 + 8 c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$- 6 b = - 512 + 8 c$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Divide each term in $- 6 b = - 512 + 8 c$ by $- 6$ and simplify.

Divide each term in $- 6 b = - 512 + 8 c$ by $- 6$ .

$\frac{- 6 b}{- 6} = \frac{- 512}{- 6} + \frac{8 c}{- 6}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 b}{- 6} = \frac{- 512}{- 6} + \frac{8 c}{- 6}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Divide $b$ by $1$ .

$b = \frac{- 512}{- 6} + \frac{8 c}{- 6}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$b = \frac{- 512}{- 6} + \frac{8 c}{- 6}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$b = \frac{- 512}{- 6} + \frac{8 c}{- 6}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 512$ and $- 6$ .

Factor $- 2$ out of $- 512$ .

$b = \frac{- 2 \cdot 256}{- 6} + \frac{8 c}{- 6}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Cancel the common factors.

Factor $- 2$ out of $- 6$ .

$b = \frac{- 2 \cdot 256}{- 2 \cdot 3} + \frac{8 c}{- 6}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Cancel the common factor.

$b = \frac{- 2 \cdot 256}{- 2 \cdot 3} + \frac{8 c}{- 6}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Rewrite the expression.

$b = \frac{256}{3} + \frac{8 c}{- 6}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$b = \frac{256}{3} + \frac{8 c}{- 6}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$b = \frac{256}{3} + \frac{8 c}{- 6}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Cancel the common factor of $8$ and $- 6$ .

Factor $2$ out of $8 c$ .

$b = \frac{256}{3} + \frac{2 (4 c)}{- 6}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Cancel the common factors.

Factor $2$ out of $- 6$ .

$b = \frac{256}{3} + \frac{2 (4 c)}{2 (- 3)}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Cancel the common factor.

$b = \frac{256}{3} + \frac{2 (4 c)}{2 \cdot - 3}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Rewrite the expression.

$b = \frac{256}{3} + \frac{4 c}{- 3}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$b = \frac{256}{3} + \frac{4 c}{- 3}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$b = \frac{256}{3} + \frac{4 c}{- 3}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Move the negative in front of the fraction.

$b = \frac{256}{3} - \frac{4 c}{3}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$72 = 264 - 2 b - 3 c$

$a = 66 - b - c$

Replace all occurrences of $b$ with $\frac{256}{3} - \frac{4 c}{3}$ in each equation.

Replace all occurrences of $b$ in $72 = 264 - 2 b - 3 c$ with $\frac{256}{3} - \frac{4 c}{3}$ .

$72 = 264 - 2 (\frac{256}{3} - \frac{4 c}{3}) - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Simplify the right side.

Simplify $264 - 2 (\frac{256}{3} - \frac{4 c}{3}) - 3 c$ .

Simplify each term.

Apply the distributive property.

$72 = 264 - 2 (\frac{256}{3}) - 2 (- \frac{4 c}{3}) - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Multiply $- 2 (\frac{256}{3})$ .

Combine $- 2$ and $\frac{256}{3}$ .

$72 = 264 + \frac{- 2 \cdot 256}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Multiply $- 2$ by $256$ .

$72 = 264 + \frac{- 512}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

$72 = 264 + \frac{- 512}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Multiply $- 2 (- \frac{4 c}{3})$ .

Multiply $- 1$ by $- 2$ .

$72 = 264 + \frac{- 512}{3} + 2 (\frac{4 c}{3}) - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Combine $2$ and $\frac{4 c}{3}$ .

$72 = 264 + \frac{- 512}{3} + \frac{2 (4 c)}{3} - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Multiply $4$ by $2$ .

$72 = 264 + \frac{- 512}{3} + \frac{8 c}{3} - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

$72 = 264 + \frac{- 512}{3} + \frac{8 c}{3} - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Move the negative in front of the fraction.

$72 = 264 - \frac{512}{3} + \frac{8 c}{3} - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

$72 = 264 - \frac{512}{3} + \frac{8 c}{3} - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

To write $264$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$72 = 264 \cdot \frac{3}{3} - \frac{512}{3} + \frac{8 c}{3} - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Combine $264$ and $\frac{3}{3}$ .

$72 = \frac{264 \cdot 3}{3} - \frac{512}{3} + \frac{8 c}{3} - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Combine the numerators over the common denominator.

$72 = \frac{264 \cdot 3 - 512}{3} + \frac{8 c}{3} - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Simplify the numerator.

Multiply $264$ by $3$ .

$72 = \frac{792 - 512}{3} + \frac{8 c}{3} - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Subtract $512$ from $792$ .

$72 = \frac{280}{3} + \frac{8 c}{3} - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

$72 = \frac{280}{3} + \frac{8 c}{3} - 3 c$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

To write $- 3 c$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$72 = \frac{280}{3} + \frac{8 c}{3} - 3 c \cdot \frac{3}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Combine $- 3 c$ and $\frac{3}{3}$ .

$72 = \frac{280}{3} + \frac{8 c}{3} + \frac{- 3 c \cdot 3}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Combine the numerators over the common denominator.

$72 = \frac{280}{3} + \frac{8 c - 3 c \cdot 3}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Combine the numerators over the common denominator.

$72 = \frac{280 + 8 c - 3 c \cdot 3}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Multiply $3$ by $- 3$ .

$72 = \frac{280 + 8 c - 9 c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Subtract $9 c$ from $8 c$ .

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - b - c$

Replace all occurrences of $b$ in $a = 66 - b - c$ with $\frac{256}{3} - \frac{4 c}{3}$ .

$a = 66 - (\frac{256}{3} - \frac{4 c}{3}) - c$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Simplify the right side.

Simplify $66 - (\frac{256}{3} - \frac{4 c}{3}) - c$ .

Simplify each term.

Apply the distributive property.

$a = 66 - \frac{256}{3} + \frac{4 c}{3} - c$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Multiply $- - \frac{4 c}{3}$ .

Multiply $- 1$ by $- 1$ .

$a = 66 - \frac{256}{3} + 1 (\frac{4 c}{3}) - c$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Multiply $\frac{4 c}{3}$ by $1$ .

$a = 66 - \frac{256}{3} + \frac{4 c}{3} - c$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - \frac{256}{3} + \frac{4 c}{3} - c$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 66 - \frac{256}{3} + \frac{4 c}{3} - c$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

To write $66$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a = 66 \cdot \frac{3}{3} - \frac{256}{3} + \frac{4 c}{3} - c$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Combine $66$ and $\frac{3}{3}$ .

$a = \frac{66 \cdot 3}{3} - \frac{256}{3} + \frac{4 c}{3} - c$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Combine the numerators over the common denominator.

$a = \frac{66 \cdot 3 - 256}{3} + \frac{4 c}{3} - c$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Simplify the numerator.

Multiply $66$ by $3$ .

$a = \frac{198 - 256}{3} + \frac{4 c}{3} - c$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Subtract $256$ from $198$ .

$a = \frac{- 58}{3} + \frac{4 c}{3} - c$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = \frac{- 58}{3} + \frac{4 c}{3} - c$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Move the negative in front of the fraction.

$a = - \frac{58}{3} + \frac{4 c}{3} - c$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

To write $- c$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a = - \frac{58}{3} + \frac{4 c}{3} - c \cdot \frac{3}{3}$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Combine $- c$ and $\frac{3}{3}$ .

$a = - \frac{58}{3} + \frac{4 c}{3} + \frac{- c \cdot 3}{3}$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Combine the numerators over the common denominator.

$a = - \frac{58}{3} + \frac{4 c - c \cdot 3}{3}$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Combine the numerators over the common denominator.

$a = \frac{- 58 + 4 c - c \cdot 3}{3}$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Multiply $3$ by $- 1$ .

$a = \frac{- 58 + 4 c - 3 c}{3}$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Subtract $3 c$ from $4 c$ .

$a = \frac{- 58 + c}{3}$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Rewrite $- 58$ as $- 1 (58)$ .

$a = \frac{- 1 \cdot 58 + c}{3}$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Factor $- 1$ out of $c$ .

$a = \frac{- 1 \cdot 58 - 1 (- c)}{3}$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Factor $- 1$ out of $- 1 (58) - 1 (- c)$ .

$a = \frac{- 1 (58 - c)}{3}$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Move the negative in front of the fraction.

$a = - \frac{58 - c}{3}$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = - \frac{58 - c}{3}$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = - \frac{58 - c}{3}$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = - \frac{58 - c}{3}$

$72 = \frac{280 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Solve for $c$ in $72 = \frac{280 - c}{3}$ .

Rewrite the equation as $\frac{280 - c}{3} = 72$ .

$\frac{280 - c}{3} = 72$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Multiply both sides by $3$ .

$\frac{280 - c}{3} \cdot 3 = 72 \cdot 3$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Simplify.

Simplify the left side.

Simplify $\frac{280 - c}{3} \cdot 3$ .

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{280 - c}{3} \cdot 3 = 72 \cdot 3$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Rewrite the expression.

$280 - c = 72 \cdot 3$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$280 - c = 72 \cdot 3$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Reorder $280$ and $- c$ .

$- c + 280 = 72 \cdot 3$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$- c + 280 = 72 \cdot 3$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$- c + 280 = 72 \cdot 3$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Simplify the right side.

Multiply $72$ by $3$ .

$- c + 280 = 216$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$- c + 280 = 216$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$- c + 280 = 216$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Solve for $c$ .

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

Subtract $280$ from both sides of the equation.

$- c = 216 - 280$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Subtract $280$ from $216$ .

$- c = - 64$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$- c = - 64$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Divide each term in $- c = - 64$ by $- 1$ and simplify.

Divide each term in $- c = - 64$ by $- 1$ .

$\frac{- c}{- 1} = \frac{- 64}{- 1}$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{c}{1} = \frac{- 64}{- 1}$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Divide $c$ by $1$ .

$c = \frac{- 64}{- 1}$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$c = \frac{- 64}{- 1}$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Simplify the right side.

Divide $- 64$ by $- 1$ .

$c = 64$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$c = 64$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$c = 64$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$c = 64$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

$c = 64$

$a = - \frac{58 - c}{3}$

$b = \frac{256}{3} - \frac{4 c}{3}$

Replace all occurrences of $c$ with $64$ in each equation.

Replace all occurrences of $c$ in $a = - \frac{58 - c}{3}$ with $64$ .

$a = - \frac{58 - (64)}{3}$

$c = 64$

$b = \frac{256}{3} - \frac{4 c}{3}$

Simplify the right side.

Simplify $- \frac{58 - (64)}{3}$ .

Simplify the numerator.

Multiply $- 1$ by $64$ .

$a = - \frac{58 - 64}{3}$

$c = 64$

$b = \frac{256}{3} - \frac{4 c}{3}$

Subtract $64$ from $58$ .

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

$c = 64$

$b = \frac{256}{3} - \frac{4 c}{3}$

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

$c = 64$

$b = \frac{256}{3} - \frac{4 c}{3}$

Simplify the expression.

Divide $- 6$ by $3$ .

$a = 2$

$c = 64$

$b = \frac{256}{3} - \frac{4 c}{3}$

Multiply $- 1$ by $- 2$ .

$a = 2$

$c = 64$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 2$

$c = 64$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 2$

$c = 64$

$b = \frac{256}{3} - \frac{4 c}{3}$

$a = 2$

$c = 64$

$b = \frac{256}{3} - \frac{4 c}{3}$

Replace all occurrences of $c$ in $b = \frac{256}{3} - \frac{4 c}{3}$ with $64$ .

$b = \frac{256}{3} - \frac{4 (64)}{3}$

$a = 2$

$c = 64$

Simplify the right side.

Simplify $\frac{256}{3} - \frac{4 (64)}{3}$ .

Combine the numerators over the common denominator.

$b = \frac{256 - 4 \cdot 64}{3}$

$a = 2$

$c = 64$

Simplify the expression.

Multiply $- 4$ by $64$ .

$b = \frac{256 - 256}{3}$

$a = 2$

$c = 64$

Subtract $256$ from $256$ .

$b = \frac{0}{3}$

$a = 2$

$c = 64$

Divide $0$ by $3$ .

$b = 0$

$a = 2$

$c = 64$

$b = 0$

$a = 2$

$c = 64$

$b = 0$

$a = 2$

$c = 64$

$b = 0$

$a = 2$

$c = 64$

$b = 0$

$a = 2$

$c = 64$

List all of the solutions.

$b = 0, a = 2, c = 64$

Calculate the value of $y$ using each $x$ value in the table and compare this value to the given $y$ value in the table.

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 2$ , $b = 0$ , $c = 64$ , and $x = 1$ .

Simplify each term.

One to any power is one.

$y = 2 \cdot 1 + (0) \cdot (1) + 64$

Multiply $2$ by $1$ .

$y = 2 + (0) \cdot (1) + 64$

Multiply $0$ by $1$ .

$y = 2 + 0 + 64$

Simplify by adding numbers.

Add $2$ and $0$ .

$y = 2 + 64$

Add $2$ and $64$ .

$y = 66$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = 66$ and $y = 66$ .

$66 = 66$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 2$ , $b = 0$ , $c = 64$ , and $x = 2$ .

Simplify each term.

Multiply $2$ by ${(2)}^{2}$ by adding the exponents.

Multiply $2$ by ${(2)}^{2}$ .

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

$y = 2 \cdot {(2)}^{2} + (0) \cdot (2) + 64$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = 2^{1 + 2} + (0) \cdot (2) + 64$

Add $1$ and $2$ .

$y = 2^{3} + (0) \cdot (2) + 64$

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

$y = 8 + (0) \cdot (2) + 64$

Multiply $0$ by $2$ .

$y = 8 + 0 + 64$

Simplify by adding numbers.

Add $8$ and $0$ .

$y = 8 + 64$

Add $8$ and $64$ .

$y = 72$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 72$ and $y = 72$ .

$72 = 72$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 2$ , $b = 0$ , $c = 64$ , and $x = 3$ .

Simplify each term.

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

$y = 2 \cdot 9 + (0) \cdot (3) + 64$

Multiply $2$ by $9$ .

$y = 18 + (0) \cdot (3) + 64$

Multiply $0$ by $3$ .

$y = 18 + 0 + 64$

Simplify by adding numbers.

Add $18$ and $0$ .

$y = 18 + 64$

Add $18$ and $64$ .

$y = 82$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = 82$ and $y = 82$ .

$82 = 82$

Since $y = y$ for the corresponding $x$ values, the function is quadratic.

The function is quadratic

Since all $y = y$ , the function is quadratic and follows the form $y = 2 x^{2} + 64$ .

$y = 2 x^{2} + 64$

Find the Function Rule table[[x,y],[1,66],[2,72],[3,82]]

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