Find the Function Rule table[[x,y],[2,9],[4,81],[6,729]]

$\begin{array}{c} x & y \\ 2 & 9 \\ 4 & 81 \\ 6 & 729 \end{array}$

Check if the function rule is linear.

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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 $y = a x + b$ .

$9 = a (2) + b 81 = a (4) + b 729 = a (6) + b$

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

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Simplify each equation.

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

$9 = 2 a + b$

$81 = a (4) + b$

$729 = a (6) + b$

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

$9 = 2 a + b$

$81 = 4 a + b$

$729 = a (6) + b$

Move $6$ to the left of $a$ .

$9 = 2 a + b$

$81 = 4 a + b$

$729 = 6 a + b$

$9 = 2 a + b$

$81 = 4 a + b$

$729 = 6 a + b$

Solve for $b$ in the first equation.

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Rewrite the equation as $2 a + b = 9$ .

$2 a + b = 9$

$81 = 4 a + b$

$729 = 6 a + b$

Subtract $2 a$ from both sides of the equation.

$b = 9 - 2 a$

$81 = 4 a + b$

$729 = 6 a + b$

$b = 9 - 2 a$

$81 = 4 a + b$

$729 = 6 a + b$

Replace all occurrences of $b$ with $9 - 2 a$ in each equation.

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Replace all occurrences of $b$ in $81 = 4 a + b$ with $9 - 2 a$ .

$b = 9 - 2 a$

$81 = 4 a + 9 - 2 a$

$729 = 6 a + b$

Replace all occurrences of $b$ in $729 = 6 a + b$ with $9 - 2 a$ .

$b = 9 - 2 a$

$81 = 4 a + 9 - 2 a$

$729 = 6 a + 9 - 2 a$

$b = 9 - 2 a$

$81 = 4 a + 9 - 2 a$

$729 = 6 a + 9 - 2 a$

Simplify each equation.

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Remove parentheses.

$b = 9 - 2 a$

$81 = 4 a + 9 - 2 a$

$729 = 6 a + 9 - 2 a$

Subtract $2 a$ from $4 a$ .

$b = 9 - 2 a$

$81 = 2 a + 9$

$729 = 6 a + 9 - 2 a$

Remove parentheses.

$b = 9 - 2 a$

$81 = 2 a + 9$

$729 = 6 a + 9 - 2 a$

Subtract $2 a$ from $6 a$ .

$b = 9 - 2 a$

$81 = 2 a + 9$

$729 = 4 a + 9$

$b = 9 - 2 a$

$81 = 2 a + 9$

$729 = 4 a + 9$

Solve for $a$ in the second equation.

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Rewrite the equation as $2 a + 9 = 81$ .

$b = 9 - 2 a$

$2 a + 9 = 81$

$729 = 4 a + 9$

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

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Subtract $9$ from both sides of the equation.

$b = 9 - 2 a$

$2 a = 81 - 9$

$729 = 4 a + 9$

Subtract $9$ from $81$ .

$b = 9 - 2 a$

$2 a = 72$

$729 = 4 a + 9$

$b = 9 - 2 a$

$2 a = 72$

$729 = 4 a + 9$

Divide each term by $2$ and simplify.

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Divide each term in $2 a = 72$ by $2$ .

$b = 9 - 2 a$

$\frac{2 a}{2} = \frac{72}{2}$

$729 = 4 a + 9$

Cancel the common factor of $2$ .

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Cancel the common factor.

$b = 9 - 2 a$

$\frac{2 a}{2} = \frac{72}{2}$

$729 = 4 a + 9$

Divide $a$ by $1$ .

$b = 9 - 2 a$

$a = \frac{72}{2}$

$729 = 4 a + 9$

$b = 9 - 2 a$

$a = \frac{72}{2}$

$729 = 4 a + 9$

Divide $72$ by $2$ .

$b = 9 - 2 a$

$a = 36$

$729 = 4 a + 9$

$b = 9 - 2 a$

$a = 36$

$729 = 4 a + 9$

$b = 9 - 2 a$

$a = 36$

$729 = 4 a + 9$

Replace all occurrences of $a$ with $36$ in each equation.

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Replace all occurrences of $a$ in $b = 9 - 2 a$ with $36$ .

$b = 9 - 2 \cdot 36$

$a = 36$

$729 = 4 a + 9$

Replace all occurrences of $a$ in $729 = 4 a + 9$ with $36$ .

$b = 9 - 2 \cdot 36$

$a = 36$

$729 = 4 (36) + 9$

$b = 9 - 2 \cdot 36$

$a = 36$

$729 = 4 (36) + 9$

Simplify.

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Simplify $9 - 2 \cdot 36$ .

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Multiply $- 2$ by $36$ .

$b = 9 - 72$

$a = 36$

$729 = 4 (36) + 9$

Subtract $72$ from $9$ .

$b = - 63$

$a = 36$

$729 = 4 (36) + 9$

$b = - 63$

$a = 36$

$729 = 4 (36) + 9$

Simplify $4 (36) + 9$ .

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Multiply $4$ by $36$ .

$b = - 63$

$a = 36$

$729 = 144 + 9$

Add $144$ and $9$ .

$b = - 63$

$a = 36$

$729 = 153$

$b = - 63$

$a = 36$

$729 = 153$

$b = - 63$

$a = 36$

$729 = 153$

Since $729 \neq 153$ , there are no solutions.

$b = - 63$

$a = 36$

No solution

$b = - 63$

$a = 36$

No solution

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

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Calculate the value of $y$ when $a = 36$ , $b = - 63$ , and $x = 2$ .

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Multiply $36$ by $2$ .

$y = 72 - 63$

Subtract $63$ from $72$ .

$y = 9$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 9$ and $y = 9$ .

$9 = 9$

Calculate the value of $y$ when $a = 36$ , $b = - 63$ , and $x = 4$ .

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Multiply $36$ by $4$ .

$y = 144 - 63$

Subtract $63$ from $144$ .

$y = 81$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 4$ . This check passes since $y = 81$ and $y = 81$ .

$81 = 81$

Calculate the value of $y$ when $a = 36$ , $b = - 63$ , and $x = 6$ .

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Multiply $36$ by $6$ .

$y = 216 - 63$

Subtract $63$ from $216$ .

$y = 153$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 6$ . This check does not pass, since $y = 153$ and $y = 729$ . The function rule can’t be linear.

$153 \neq 729$

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.

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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 $c$ .

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Simplify each equation.

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Simplify each term.

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

$9 = a \cdot 4 + b (2) + c$

$81 = a {(4)}^{2} + b (4) + c$

$729 = a {(6)}^{2} + b (6) + c$

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

$9 = 4 \cdot a + b (2) + c$

$81 = a {(4)}^{2} + b (4) + c$

$729 = a {(6)}^{2} + b (6) + c$

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

$9 = 4 a + 2 b + c$

$81 = a {(4)}^{2} + b (4) + c$

$729 = a {(6)}^{2} + b (6) + c$

$9 = 4 a + 2 b + c$

$81 = a {(4)}^{2} + b (4) + c$

$729 = a {(6)}^{2} + b (6) + c$

Simplify each term.

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

$9 = 4 a + 2 b + c$

$81 = a \cdot 16 + b (4) + c$

$729 = a {(6)}^{2} + b (6) + c$

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

$9 = 4 a + 2 b + c$

$81 = 16 \cdot a + b (4) + c$

$729 = a {(6)}^{2} + b (6) + c$

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

$9 = 4 a + 2 b + c$

$81 = 16 a + 4 b + c$

$729 = a {(6)}^{2} + b (6) + c$

$9 = 4 a + 2 b + c$

$81 = 16 a + 4 b + c$

$729 = a {(6)}^{2} + b (6) + c$

Simplify each term.

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Raise $6$ to the power of $2$ .

$9 = 4 a + 2 b + c$

$81 = 16 a + 4 b + c$

$729 = a \cdot 36 + b (6) + c$

Move $36$ to the left of $a$ .

$9 = 4 a + 2 b + c$

$81 = 16 a + 4 b + c$

$729 = 36 \cdot a + b (6) + c$

Move $6$ to the left of $b$ .

$9 = 4 a + 2 b + c$

$81 = 16 a + 4 b + c$

$729 = 36 a + 6 b + c$

$9 = 4 a + 2 b + c$

$81 = 16 a + 4 b + c$

$729 = 36 a + 6 b + c$

$9 = 4 a + 2 b + c$

$81 = 16 a + 4 b + c$

$729 = 36 a + 6 b + c$

Solve for $c$ in the first equation.

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Rewrite the equation as $4 a + 2 b + c = 9$ .

$4 a + 2 b + c = 9$

$81 = 16 a + 4 b + c$

$729 = 36 a + 6 b + c$

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

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

$2 b + c = 9 - 4 a$

$81 = 16 a + 4 b + c$

$729 = 36 a + 6 b + c$

Subtract $2 b$ from both sides of the equation.

$c = 9 - 4 a - 2 b$

$81 = 16 a + 4 b + c$

$729 = 36 a + 6 b + c$

$c = 9 - 4 a - 2 b$

$81 = 16 a + 4 b + c$

$729 = 36 a + 6 b + c$

$c = 9 - 4 a - 2 b$

$81 = 16 a + 4 b + c$

$729 = 36 a + 6 b + c$

Replace all occurrences of $c$ with $9 - 4 a - 2 b$ in each equation.

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Replace all occurrences of $c$ in $81 = 16 a + 4 b + c$ with $9 - 4 a - 2 b$ .

$c = 9 - 4 a - 2 b$

$81 = 16 a + 4 b + 9 - 4 a - 2 b$

$729 = 36 a + 6 b + c$

Replace all occurrences of $c$ in $729 = 36 a + 6 b + c$ with $9 - 4 a - 2 b$ .

$c = 9 - 4 a - 2 b$

$81 = 16 a + 4 b + 9 - 4 a - 2 b$

$729 = 36 a + 6 b + 9 - 4 a - 2 b$

$c = 9 - 4 a - 2 b$

$81 = 16 a + 4 b + 9 - 4 a - 2 b$

$729 = 36 a + 6 b + 9 - 4 a - 2 b$

Simplify each equation.

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Remove parentheses.

$c = 9 - 4 a - 2 b$

$81 = 16 a + 4 b + 9 - 4 a - 2 b$

$729 = 36 a + 6 b + 9 - 4 a - 2 b$

Simplify $16 a + 4 b + 9 - 4 a - 2 b$ .

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Subtract $4 a$ from $16 a$ .

$c = 9 - 4 a - 2 b$

$81 = 12 a + 4 b + 9 - 2 b$

$729 = 36 a + 6 b + 9 - 4 a - 2 b$

Subtract $2 b$ from $4 b$ .

$c = 9 - 4 a - 2 b$

$81 = 12 a + 2 b + 9$

$729 = 36 a + 6 b + 9 - 4 a - 2 b$

$c = 9 - 4 a - 2 b$

$81 = 12 a + 2 b + 9$

$729 = 36 a + 6 b + 9 - 4 a - 2 b$

Remove parentheses.

$c = 9 - 4 a - 2 b$

$81 = 12 a + 2 b + 9$

$729 = 36 a + 6 b + 9 - 4 a - 2 b$

Simplify $36 a + 6 b + 9 - 4 a - 2 b$ .

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Subtract $4 a$ from $36 a$ .

$c = 9 - 4 a - 2 b$

$81 = 12 a + 2 b + 9$

$729 = 32 a + 6 b + 9 - 2 b$

Subtract $2 b$ from $6 b$ .

$c = 9 - 4 a - 2 b$

$81 = 12 a + 2 b + 9$

$729 = 32 a + 4 b + 9$

$c = 9 - 4 a - 2 b$

$81 = 12 a + 2 b + 9$

$729 = 32 a + 4 b + 9$

$c = 9 - 4 a - 2 b$

$81 = 12 a + 2 b + 9$

$729 = 32 a + 4 b + 9$

Solve for $a$ in the second equation.

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Rewrite the equation as $12 a + 2 b + 9 = 81$ .

$c = 9 - 4 a - 2 b$

$12 a + 2 b + 9 = 81$

$729 = 32 a + 4 b + 9$

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

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Subtract $2 b$ from both sides of the equation.

$c = 9 - 4 a - 2 b$

$12 a + 9 = 81 - 2 b$

$729 = 32 a + 4 b + 9$

Subtract $9$ from both sides of the equation.

$c = 9 - 4 a - 2 b$

$12 a = 81 - 2 b - 9$

$729 = 32 a + 4 b + 9$

Subtract $9$ from $81$ .

$c = 9 - 4 a - 2 b$

$12 a = - 2 b + 72$

$729 = 32 a + 4 b + 9$

$c = 9 - 4 a - 2 b$

$12 a = - 2 b + 72$

$729 = 32 a + 4 b + 9$

Divide each term by $12$ and simplify.

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Divide each term in $12 a = - 2 b + 72$ by $12$ .

$c = 9 - 4 a - 2 b$

$\frac{12 a}{12} = \frac{- 2 b}{12} + \frac{72}{12}$

$729 = 32 a + 4 b + 9$

Cancel the common factor of $12$ .

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Cancel the common factor.

$c = 9 - 4 a - 2 b$

$\frac{12 a}{12} = \frac{- 2 b}{12} + \frac{72}{12}$

$729 = 32 a + 4 b + 9$

Divide $a$ by $1$ .

$c = 9 - 4 a - 2 b$

$a = \frac{- 2 b}{12} + \frac{72}{12}$

$729 = 32 a + 4 b + 9$

$c = 9 - 4 a - 2 b$

$a = \frac{- 2 b}{12} + \frac{72}{12}$

$729 = 32 a + 4 b + 9$

Simplify each term.

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Cancel the common factor of $- 2$ and $12$ .

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Factor $2$ out of $- 2 b$ .

$c = 9 - 4 a - 2 b$

$a = \frac{2 (- b)}{12} + \frac{72}{12}$

$729 = 32 a + 4 b + 9$

Cancel the common factors.

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Factor $2$ out of $12$ .

$c = 9 - 4 a - 2 b$

$a = \frac{2 (- b)}{2 (6)} + \frac{72}{12}$

$729 = 32 a + 4 b + 9$

Cancel the common factor.

$c = 9 - 4 a - 2 b$

$a = \frac{2 (- b)}{2 \cdot 6} + \frac{72}{12}$

$729 = 32 a + 4 b + 9$

Rewrite the expression.

$c = 9 - 4 a - 2 b$

$a = \frac{- b}{6} + \frac{72}{12}$

$729 = 32 a + 4 b + 9$

$c = 9 - 4 a - 2 b$

$a = \frac{- b}{6} + \frac{72}{12}$

$729 = 32 a + 4 b + 9$

$c = 9 - 4 a - 2 b$

$a = \frac{- b}{6} + \frac{72}{12}$

$729 = 32 a + 4 b + 9$

Move the negative in front of the fraction.

$c = 9 - 4 a - 2 b$

$a = - \frac{b}{6} + \frac{72}{12}$

$729 = 32 a + 4 b + 9$

Divide $72$ by $12$ .

$c = 9 - 4 a - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 a + 4 b + 9$

$c = 9 - 4 a - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 a + 4 b + 9$

$c = 9 - 4 a - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 a + 4 b + 9$

$c = 9 - 4 a - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 a + 4 b + 9$

Replace all occurrences of $a$ with $- \frac{b}{6} + 6$ in each equation.

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Replace all occurrences of $a$ in $c = 9 - 4 a - 2 b$ with $- \frac{b}{6} + 6$ .

$c = 9 - 4 (- \frac{b}{6} + 6) - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 a + 4 b + 9$

Replace all occurrences of $a$ in $729 = 32 a + 4 b + 9$ with $- \frac{b}{6} + 6$ .

$c = 9 - 4 (- \frac{b}{6} + 6) - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

$c = 9 - 4 (- \frac{b}{6} + 6) - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Simplify each equation.

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Simplify $9 - 4 (- \frac{b}{6} + 6) - 2 b$ .

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Simplify each term.

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Apply the distributive property.

$c = 9 - 4 (- \frac{b}{6}) - 4 \cdot 6 - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{b}{6}$ into the numerator.

$c = 9 - 4 \frac{- b}{6} - 4 \cdot 6 - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Factor $2$ out of $- 4$ .

$c = 9 + 2 (- 2) (\frac{- b}{6}) - 4 \cdot 6 - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Factor $2$ out of $6$ .

$c = 9 + 2 \cdot (- 2 \frac{- b}{2 \cdot 3}) - 4 \cdot 6 - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Cancel the common factor.

$c = 9 + 2 \cdot (- 2 \frac{- b}{2 \cdot 3}) - 4 \cdot 6 - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Rewrite the expression.

$c = 9 - 2 \frac{- b}{3} - 4 \cdot 6 - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

$c = 9 - 2 \frac{- b}{3} - 4 \cdot 6 - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

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

$c = 9 + \frac{- 2 (- b)}{3} - 4 \cdot 6 - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Multiply $- 1$ by $- 2$ .

$c = 9 + \frac{2 b}{3} - 4 \cdot 6 - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Multiply $- 4$ by $6$ .

$c = 9 + \frac{2 b}{3} - 24 - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

$c = 9 + \frac{2 b}{3} - 24 - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Subtract $24$ from $9$ .

$c = \frac{2 b}{3} - 15 - 2 b$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

To write $- 2 b$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$c = \frac{2 b}{3} - 2 b \cdot \frac{3}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Simplify terms.

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

$c = \frac{2 b}{3} + \frac{- 2 b \cdot 3}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Combine the numerators over the common denominator.

$c = \frac{2 b - 2 b \cdot 3}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

$c = \frac{2 b - 2 b \cdot 3}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Simplify each term.

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Simplify the numerator.

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Factor $2 b$ out of $2 b - 2 b \cdot 3$ .

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Factor $2 b$ out of $2 b$ .

$c = \frac{2 b (1) - 2 b \cdot 3}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Factor $2 b$ out of $- 2 b \cdot 3$ .

$c = \frac{2 b (1) + 2 b (- 1 \cdot 3)}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Factor $2 b$ out of $2 b (1) + 2 b (- 1 \cdot 3)$ .

$c = \frac{2 b (1 - 1 \cdot 3)}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

$c = \frac{2 b (1 - 1 \cdot 3)}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Multiply $- 1$ by $3$ .

$c = \frac{2 b (1 - 3)}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Subtract $3$ from $1$ .

$c = \frac{2 b \cdot - 2}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

$c = \frac{2 b \cdot - 2}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Multiply $- 2$ by $2$ .

$c = \frac{- 4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Move the negative in front of the fraction.

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6} + 6) + 4 b + 9$

Simplify $32 (- \frac{b}{6} + 6) + 4 b + 9$ .

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Simplify each term.

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Apply the distributive property.

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (- \frac{b}{6}) + 32 \cdot 6 + 4 b + 9$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{b}{6}$ into the numerator.

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 32 (\frac{- b}{6}) + 32 \cdot 6 + 4 b + 9$

Factor $2$ out of $32$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 2 (16) (\frac{- b}{6}) + 32 \cdot 6 + 4 b + 9$

Factor $2$ out of $6$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 2 \cdot (16 (\frac{- b}{2 \cdot 3})) + 32 \cdot 6 + 4 b + 9$

Cancel the common factor.

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 2 \cdot (16 (\frac{- b}{2 \cdot 3})) + 32 \cdot 6 + 4 b + 9$

Rewrite the expression.

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 16 (\frac{- b}{3}) + 32 \cdot 6 + 4 b + 9$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = 16 (\frac{- b}{3}) + 32 \cdot 6 + 4 b + 9$

Combine $16$ and $\frac{- b}{3}$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{16 (- b)}{3} + 32 \cdot 6 + 4 b + 9$

Multiply $- 1$ by $16$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b}{3} + 32 \cdot 6 + 4 b + 9$

Multiply $32$ by $6$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b}{3} + 192 + 4 b + 9$

Move the negative in front of the fraction.

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = - \frac{16 b}{3} + 192 + 4 b + 9$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = - \frac{16 b}{3} + 192 + 4 b + 9$

To write $4 b$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = - \frac{16 b}{3} + 4 b \cdot \frac{3}{3} + 192 + 9$

Combine $4 b$ and $\frac{3}{3}$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = - \frac{16 b}{3} + \frac{4 b \cdot 3}{3} + 192 + 9$

Combine the numerators over the common denominator.

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b + 4 b \cdot 3}{3} + 192 + 9$

Find the common denominator.

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Write $192$ as a fraction with denominator $1$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b + 4 b \cdot 3}{3} + \frac{192}{1} + 9$

Multiply $\frac{192}{1}$ by $\frac{3}{3}$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b + 4 b \cdot 3}{3} + \frac{192}{1} \cdot \frac{3}{3} + 9$

Multiply $\frac{192}{1}$ and $\frac{3}{3}$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b + 4 b \cdot 3}{3} + \frac{192 \cdot 3}{3} + 9$

Write $9$ as a fraction with denominator $1$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b + 4 b \cdot 3}{3} + \frac{192 \cdot 3}{3} + \frac{9}{1}$

Multiply $\frac{9}{1}$ by $\frac{3}{3}$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b + 4 b \cdot 3}{3} + \frac{192 \cdot 3}{3} + \frac{9}{1} \cdot \frac{3}{3}$

Multiply $\frac{9}{1}$ and $\frac{3}{3}$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b + 4 b \cdot 3}{3} + \frac{192 \cdot 3}{3} + \frac{9 \cdot 3}{3}$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b + 4 b \cdot 3}{3} + \frac{192 \cdot 3}{3} + \frac{9 \cdot 3}{3}$

Combine fractions with similar denominators.

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b + 4 b \cdot 3 + 192 \cdot 3 + 9 \cdot 3}{3}$

Simplify the expression.

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Multiply $192$ by $3$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b + 4 b \cdot 3 + 576 + 9 \cdot 3}{3}$

Multiply $9$ by $3$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b + 4 b \cdot 3 + 576 + 27}{3}$

Add $576$ and $27$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b + 4 b \cdot 3 + 603}{3}$

Multiply $3$ by $4$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b + 12 b + 603}{3}$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 16 b + 12 b + 603}{3}$

Add $- 16 b$ and $12 b$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 4 b + 603}{3}$

Factor $- 1$ out of $- 4 b$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- (4 b) + 603}{3}$

Rewrite $603$ as $- 1 (- 603)$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- (4 b) - 1 \cdot - 603}{3}$

Factor $- 1$ out of $- (4 b) - 1 (- 603)$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- (4 b - 603)}{3}$

Simplify the expression.

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Rewrite $- (4 b - 603)$ as $- 1 (4 b - 603)$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = \frac{- 1 (4 b - 603)}{3}$

Move the negative in front of the fraction.

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = - \frac{4 b - 603}{3}$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = - \frac{4 b - 603}{3}$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = - \frac{4 b - 603}{3}$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$729 = - \frac{4 b - 603}{3}$

Solve for $b$ in the third equation.

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Rewrite the equation as $- \frac{4 b - 603}{3} = 729$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$- \frac{4 b - 603}{3} = 729$

Multiply both sides of the equation by $- 3$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$- 3 \cdot (- \frac{4 b - 603}{3}) = - 3 \cdot 729$

Simplify both sides of the equation.

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Simplify $- 3 \cdot (- \frac{4 b - 603}{3})$ .

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Cancel the common factor of $3$ .

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Move the leading negative in $- \frac{4 b - 603}{3}$ into the numerator.

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$- 3 \cdot \frac{- (4 b - 603)}{3} = - 3 \cdot 729$

Factor $3$ out of $- 3$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$3 (- 1) \cdot \frac{- (4 b - 603)}{3} = - 3 \cdot 729$

Cancel the common factor.

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$3 \cdot - 1 \cdot \frac{- (4 b - 603)}{3} = - 3 \cdot 729$

Rewrite the expression.

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$- 1 \cdot (- (4 b - 603)) = - 3 \cdot 729$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$- 1 \cdot (- (4 b - 603)) = - 3 \cdot 729$

Multiply.

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Multiply $- 1$ by $- 1$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$1 \cdot (4 b - 603) = - 3 \cdot 729$

Multiply $4 b - 603$ by $1$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$4 b - 603 = - 3 \cdot 729$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$4 b - 603 = - 3 \cdot 729$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$4 b - 603 = - 3 \cdot 729$

Multiply $- 3$ by $729$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$4 b - 603 = - 2187$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$4 b - 603 = - 2187$

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

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Add $603$ to both sides of the equation.

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$4 b = - 2187 + 603$

Add $- 2187$ and $603$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$4 b = - 1584$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$4 b = - 1584$

Divide each term by $4$ and simplify.

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Divide each term in $4 b = - 1584$ by $4$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$\frac{4 b}{4} = \frac{- 1584}{4}$

Cancel the common factor of $4$ .

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Cancel the common factor.

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$\frac{4 b}{4} = \frac{- 1584}{4}$

Divide $b$ by $1$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$b = \frac{- 1584}{4}$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$b = \frac{- 1584}{4}$

Divide $- 1584$ by $4$ .

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$b = - 396$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$b = - 396$

$c = - \frac{4 b}{3} - 15$

$a = - \frac{b}{6} + 6$

$b = - 396$

Replace all occurrences of $b$ with $- 396$ in each equation.

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Replace all occurrences of $b$ in $c = - \frac{4 b}{3} - 15$ with $- 396$ .

$c = - \frac{4 (- 396)}{3} - 15$

$a = - \frac{b}{6} + 6$

$b = - 396$

Replace all occurrences of $b$ in $a = - \frac{b}{6} + 6$ with $- 396$ .

$c = - \frac{4 (- 396)}{3} - 15$

$a = - \frac{- 396}{6} + 6$

$b = - 396$

$c = - \frac{4 (- 396)}{3} - 15$

$a = - \frac{- 396}{6} + 6$

$b = - 396$

Simplify.

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Simplify $- \frac{4 (- 396)}{3} - 15$ .

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Simplify each term.

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Cancel the common factor of $- 396$ and $3$ .

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Factor $3$ out of $4 (- 396)$ .

$c = - \frac{3 (4 \cdot (- 132))}{3} - 15$

$a = - \frac{- 396}{6} + 6$

$b = - 396$

Cancel the common factors.

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Factor $3$ out of $3$ .

$c = - \frac{3 (4 \cdot (- 132))}{3 (1)} - 15$

$a = - \frac{- 396}{6} + 6$

$b = - 396$

Cancel the common factor.

$c = - \frac{3 (4 \cdot (- 132))}{3 \cdot 1} - 15$

$a = - \frac{- 396}{6} + 6$

$b = - 396$

Rewrite the expression.

$c = - \frac{4 \cdot (- 132)}{1} - 15$

$a = - \frac{- 396}{6} + 6$

$b = - 396$

Divide $4 \cdot (- 132)$ by $1$ .

$c = - (4 \cdot (- 132)) - 15$

$a = - \frac{- 396}{6} + 6$

$b = - 396$

$c = - (4 \cdot (- 132)) - 15$

$a = - \frac{- 396}{6} + 6$

$b = - 396$

$c = - (4 \cdot (- 132)) - 15$

$a = - \frac{- 396}{6} + 6$

$b = - 396$

Multiply $4$ by $- 132$ .

$c = 528 - 15$

$a = - \frac{- 396}{6} + 6$

$b = - 396$

Multiply $- 1$ by $- 528$ .

$c = 528 - 15$

$a = - \frac{- 396}{6} + 6$

$b = - 396$

$c = 528 - 15$

$a = - \frac{- 396}{6} + 6$

$b = - 396$

Subtract $15$ from $528$ .

$c = 513$

$a = - \frac{- 396}{6} + 6$

$b = - 396$

$c = 513$

$a = - \frac{- 396}{6} + 6$

$b = - 396$

Simplify $- \frac{- 396}{6} + 6$ .

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Simplify each term.

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Divide $- 396$ by $6$ .

$c = 513$

$a = 66 + 6$

$b = - 396$

Multiply $- 1$ by $- 66$ .

$c = 513$

$a = 66 + 6$

$b = - 396$

$c = 513$

$a = 66 + 6$

$b = - 396$

Add $66$ and $6$ .

$c = 513$

$a = 72$

$b = - 396$

$c = 513$

$a = 72$

$b = - 396$

$c = 513$

$a = 72$

$b = - 396$

$c = 513$

$a = 72$

$b = - 396$

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

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Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 72$ , $b = - 396$ , $c = 513$ , and $x = 2$ .

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Simplify each term.

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

$y = 72 \cdot 4 + (- 396) \cdot (2) + 513$

Multiply $72$ by $4$ .

$y = 288 + (- 396) \cdot (2) + 513$

Multiply $- 396$ by $2$ .

$y = 288 - 792 + 513$

Simplify by adding and subtracting.

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Subtract $792$ from $288$ .

$y = - 504 + 513$

Add $- 504$ and $513$ .

$y = 9$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 9$ and $y = 9$ .

$9 = 9$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 72$ , $b = - 396$ , $c = 513$ , and $x = 4$ .

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Simplify each term.

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

$y = 72 \cdot 16 + (- 396) \cdot (4) + 513$

Multiply $72$ by $16$ .

$y = 1152 + (- 396) \cdot (4) + 513$

Multiply $- 396$ by $4$ .

$y = 1152 - 1584 + 513$

Simplify by adding and subtracting.

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Subtract $1584$ from $1152$ .

$y = - 432 + 513$

Add $- 432$ and $513$ .

$y = 81$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 4$ . This check passes since $y = 81$ and $y = 81$ .

$81 = 81$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 72$ , $b = - 396$ , $c = 513$ , and $x = 6$ .

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Simplify each term.

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Raise $6$ to the power of $2$ .

$y = 72 \cdot 36 + (- 396) \cdot (6) + 513$

Multiply $72$ by $36$ .

$y = 2592 + (- 396) \cdot (6) + 513$

Multiply $- 396$ by $6$ .

$y = 2592 - 2376 + 513$

Simplify by adding and subtracting.

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Subtract $2376$ from $2592$ .

$y = 216 + 513$

Add $216$ and $513$ .

$y = 729$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 6$ . This check passes since $y = 729$ and $y = 729$ .

$729 = 729$

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 $y = 72 x^{2} - 396 x + 513$ .

$y = 72 x^{2} - 396 x + 513$

Find the Function Rule table[[x,y],[2,9],[4,81],[6,729]]

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