Find the Function Rule table[[x,f(x)],[3,64],[4,256],[5,1024]]

$\begin{array}{c} x & f (x) \\ 3 & 64 \\ 4 & 256 \\ 5 & 1024 \end{array}$

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 $f (x) = a x + b$ .

$64 = a (3) + b 256 = a (4) + b 1024 = a (5) + b$

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

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Simplify each equation.

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

$64 = 3 a + b$

$256 = a (4) + b$

$1024 = a (5) + b$

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

$64 = 3 a + b$

$256 = 4 a + b$

$1024 = a (5) + b$

Move $5$ to the left of $a$ .

$64 = 3 a + b$

$256 = 4 a + b$

$1024 = 5 a + b$

$64 = 3 a + b$

$256 = 4 a + b$

$1024 = 5 a + b$

Solve for $b$ in the first equation.

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Rewrite the equation as $3 a + b = 64$ .

$3 a + b = 64$

$256 = 4 a + b$

$1024 = 5 a + b$

Subtract $3 a$ from both sides of the equation.

$b = 64 - 3 a$

$256 = 4 a + b$

$1024 = 5 a + b$

$b = 64 - 3 a$

$256 = 4 a + b$

$1024 = 5 a + b$

Replace all occurrences of $b$ with $64 - 3 a$ in each equation.

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Replace all occurrences of $b$ in $256 = 4 a + b$ with $64 - 3 a$ .

$b = 64 - 3 a$

$256 = 4 a + 64 - 3 a$

$1024 = 5 a + b$

Replace all occurrences of $b$ in $1024 = 5 a + b$ with $64 - 3 a$ .

$b = 64 - 3 a$

$256 = 4 a + 64 - 3 a$

$1024 = 5 a + 64 - 3 a$

$b = 64 - 3 a$

$256 = 4 a + 64 - 3 a$

$1024 = 5 a + 64 - 3 a$

Simplify each equation.

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Remove parentheses.

$b = 64 - 3 a$

$256 = 4 a + 64 - 3 a$

$1024 = 5 a + 64 - 3 a$

Subtract $3 a$ from $4 a$ .

$b = 64 - 3 a$

$256 = a + 64$

$1024 = 5 a + 64 - 3 a$

Remove parentheses.

$b = 64 - 3 a$

$256 = a + 64$

$1024 = 5 a + 64 - 3 a$

Subtract $3 a$ from $5 a$ .

$b = 64 - 3 a$

$256 = a + 64$

$1024 = 2 a + 64$

$b = 64 - 3 a$

$256 = a + 64$

$1024 = 2 a + 64$

Solve for $a$ in the second equation.

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Rewrite the equation as $a + 64 = 256$ .

$b = 64 - 3 a$

$a + 64 = 256$

$1024 = 2 a + 64$

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

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Subtract $64$ from both sides of the equation.

$b = 64 - 3 a$

$a = 256 - 64$

$1024 = 2 a + 64$

Subtract $64$ from $256$ .

$b = 64 - 3 a$

$a = 192$

$1024 = 2 a + 64$

$b = 64 - 3 a$

$a = 192$

$1024 = 2 a + 64$

$b = 64 - 3 a$

$a = 192$

$1024 = 2 a + 64$

Replace all occurrences of $a$ with $192$ in each equation.

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Replace all occurrences of $a$ in $b = 64 - 3 a$ with $192$ .

$b = 64 - 3 \cdot 192$

$a = 192$

$1024 = 2 a + 64$

Replace all occurrences of $a$ in $1024 = 2 a + 64$ with $192$ .

$b = 64 - 3 \cdot 192$

$a = 192$

$1024 = 2 (192) + 64$

$b = 64 - 3 \cdot 192$

$a = 192$

$1024 = 2 (192) + 64$

Simplify.

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Simplify $64 - 3 \cdot 192$ .

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Multiply $- 3$ by $192$ .

$b = 64 - 576$

$a = 192$

$1024 = 2 (192) + 64$

Subtract $576$ from $64$ .

$b = - 512$

$a = 192$

$1024 = 2 (192) + 64$

$b = - 512$

$a = 192$

$1024 = 2 (192) + 64$

Simplify $2 (192) + 64$ .

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Multiply $2$ by $192$ .

$b = - 512$

$a = 192$

$1024 = 384 + 64$

Add $384$ and $64$ .

$b = - 512$

$a = 192$

$1024 = 448$

$b = - 512$

$a = 192$

$1024 = 448$

$b = - 512$

$a = 192$

$1024 = 448$

Since $1024 \neq 448$ , there are no solutions.

$b = - 512$

$a = 192$

No solution

$b = - 512$

$a = 192$

No solution

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

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Calculate the value of $y$ when $a = 192$ , $b = - 512$ , and $x = 3$ .

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Multiply $192$ by $3$ .

$y = 576 - 512$

Subtract $512$ from $576$ .

$y = 64$

If the table has a linear function rule, $y = f (x)$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = 64$ and $f (x) = 64$ .

$64 = 64$

Calculate the value of $y$ when $a = 192$ , $b = - 512$ , and $x = 4$ .

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Multiply $192$ by $4$ .

$y = 768 - 512$

Subtract $512$ from $768$ .

$y = 256$

If the table has a linear function rule, $y = f (x)$ for the corresponding $x$ value, $x = 4$ . This check passes since $y = 256$ and $f (x) = 256$ .

$256 = 256$

Calculate the value of $y$ when $a = 192$ , $b = - 512$ , and $x = 5$ .

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Multiply $192$ by $5$ .

$y = 960 - 512$

Subtract $512$ from $960$ .

$y = 448$

If the table has a linear function rule, $y = f (x)$ for the corresponding $x$ value, $x = 5$ . This check does not pass, since $y = 448$ and $f (x) = 1024$ . The function rule can’t be linear.

$448 \neq 1024$

Since $y \neq f (x)$ 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 $f (x) = a x^{2} + b x + c$ .

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 $3$ to the power of $2$ .

$64 = a \cdot 9 + b (3) + c$

$256 = a {(4)}^{2} + b (4) + c$

$1024 = a {(5)}^{2} + b (5) + c$

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

$64 = 9 \cdot a + b (3) + c$

$256 = a {(4)}^{2} + b (4) + c$

$1024 = a {(5)}^{2} + b (5) + c$

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

$64 = 9 a + 3 b + c$

$256 = a {(4)}^{2} + b (4) + c$

$1024 = a {(5)}^{2} + b (5) + c$

$64 = 9 a + 3 b + c$

$256 = a {(4)}^{2} + b (4) + c$

$1024 = a {(5)}^{2} + b (5) + c$

Simplify each term.

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

$64 = 9 a + 3 b + c$

$256 = a \cdot 16 + b (4) + c$

$1024 = a {(5)}^{2} + b (5) + c$

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

$64 = 9 a + 3 b + c$

$256 = 16 \cdot a + b (4) + c$

$1024 = a {(5)}^{2} + b (5) + c$

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

$64 = 9 a + 3 b + c$

$256 = 16 a + 4 b + c$

$1024 = a {(5)}^{2} + b (5) + c$

$64 = 9 a + 3 b + c$

$256 = 16 a + 4 b + c$

$1024 = a {(5)}^{2} + b (5) + c$

Simplify each term.

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Raise $5$ to the power of $2$ .

$64 = 9 a + 3 b + c$

$256 = 16 a + 4 b + c$

$1024 = a \cdot 25 + b (5) + c$

Move $25$ to the left of $a$ .

$64 = 9 a + 3 b + c$

$256 = 16 a + 4 b + c$

$1024 = 25 \cdot a + b (5) + c$

Move $5$ to the left of $b$ .

$64 = 9 a + 3 b + c$

$256 = 16 a + 4 b + c$

$1024 = 25 a + 5 b + c$

$64 = 9 a + 3 b + c$

$256 = 16 a + 4 b + c$

$1024 = 25 a + 5 b + c$

$64 = 9 a + 3 b + c$

$256 = 16 a + 4 b + c$

$1024 = 25 a + 5 b + c$

Solve for $c$ in the first equation.

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Rewrite the equation as $9 a + 3 b + c = 64$ .

$9 a + 3 b + c = 64$

$256 = 16 a + 4 b + c$

$1024 = 25 a + 5 b + c$

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

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Subtract $9 a$ from both sides of the equation.

$3 b + c = 64 - 9 a$

$256 = 16 a + 4 b + c$

$1024 = 25 a + 5 b + c$

Subtract $3 b$ from both sides of the equation.

$c = 64 - 9 a - 3 b$

$256 = 16 a + 4 b + c$

$1024 = 25 a + 5 b + c$

$c = 64 - 9 a - 3 b$

$256 = 16 a + 4 b + c$

$1024 = 25 a + 5 b + c$

$c = 64 - 9 a - 3 b$

$256 = 16 a + 4 b + c$

$1024 = 25 a + 5 b + c$

Replace all occurrences of $c$ with $64 - 9 a - 3 b$ in each equation.

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Replace all occurrences of $c$ in $256 = 16 a + 4 b + c$ with $64 - 9 a - 3 b$ .

$c = 64 - 9 a - 3 b$

$256 = 16 a + 4 b + 64 - 9 a - 3 b$

$1024 = 25 a + 5 b + c$

Replace all occurrences of $c$ in $1024 = 25 a + 5 b + c$ with $64 - 9 a - 3 b$ .

$c = 64 - 9 a - 3 b$

$256 = 16 a + 4 b + 64 - 9 a - 3 b$

$1024 = 25 a + 5 b + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$256 = 16 a + 4 b + 64 - 9 a - 3 b$

$1024 = 25 a + 5 b + 64 - 9 a - 3 b$

Simplify each equation.

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Remove parentheses.

$c = 64 - 9 a - 3 b$

$256 = 16 a + 4 b + 64 - 9 a - 3 b$

$1024 = 25 a + 5 b + 64 - 9 a - 3 b$

Simplify $16 a + 4 b + 64 - 9 a - 3 b$ .

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Subtract $9 a$ from $16 a$ .

$c = 64 - 9 a - 3 b$

$256 = 7 a + 4 b + 64 - 3 b$

$1024 = 25 a + 5 b + 64 - 9 a - 3 b$

Subtract $3 b$ from $4 b$ .

$c = 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$1024 = 25 a + 5 b + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$1024 = 25 a + 5 b + 64 - 9 a - 3 b$

Remove parentheses.

$c = 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$1024 = 25 a + 5 b + 64 - 9 a - 3 b$

Simplify $25 a + 5 b + 64 - 9 a - 3 b$ .

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Subtract $9 a$ from $25 a$ .

$c = 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$1024 = 16 a + 5 b + 64 - 3 b$

Subtract $3 b$ from $5 b$ .

$c = 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$1024 = 16 a + 2 b + 64$

Solve for $b$ in the second equation.

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Rewrite the equation as $7 a + b + 64 = 256$ .

$c = 64 - 9 a - 3 b$

$7 a + b + 64 = 256$

$1024 = 16 a + 2 b + 64$

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

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Subtract $7 a$ from both sides of the equation.

$c = 64 - 9 a - 3 b$

$b + 64 = 256 - 7 a$

$1024 = 16 a + 2 b + 64$

Subtract $64$ from both sides of the equation.

$c = 64 - 9 a - 3 b$

$b = 256 - 7 a - 64$

$1024 = 16 a + 2 b + 64$

Subtract $64$ from $256$ .

$c = 64 - 9 a - 3 b$

$b = - 7 a + 192$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

$b = - 7 a + 192$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

$b = - 7 a + 192$

$1024 = 16 a + 2 b + 64$

Replace all occurrences of $b$ with $- 7 a + 192$ in each equation.

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Replace all occurrences of $b$ in $c = 64 - 9 a - 3 b$ with $- 7 a + 192$ .

$c = 64 - 9 a - 3 (- 7 a + 192)$

$b = - 7 a + 192$

$1024 = 16 a + 2 b + 64$

Replace all occurrences of $b$ in $1024 = 16 a + 2 b + 64$ with $- 7 a + 192$ .

$c = 64 - 9 a - 3 (- 7 a + 192)$

$b = - 7 a + 192$

$1024 = 16 a + 2 (- 7 a + 192) + 64$

$c = 64 - 9 a - 3 (- 7 a + 192)$

$b = - 7 a + 192$

$1024 = 16 a + 2 (- 7 a + 192) + 64$

Simplify each equation.

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Simplify $64 - 9 a - 3 (- 7 a + 192)$ .

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Simplify each term.

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Apply the distributive property.

$c = 64 - 9 a - 3 (- 7 a) - 3 \cdot 192$

$b = - 7 a + 192$

$1024 = 16 a + 2 (- 7 a + 192) + 64$

Multiply $- 7$ by $- 3$ .

$c = 64 - 9 a + 21 a - 3 \cdot 192$

$b = - 7 a + 192$

$1024 = 16 a + 2 (- 7 a + 192) + 64$

Multiply $- 3$ by $192$ .

$c = 64 - 9 a + 21 a - 576$

$b = - 7 a + 192$

$1024 = 16 a + 2 (- 7 a + 192) + 64$

$c = 64 - 9 a + 21 a - 576$

$b = - 7 a + 192$

$1024 = 16 a + 2 (- 7 a + 192) + 64$

Simplify by adding terms.

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Subtract $576$ from $64$ .

$c = - 9 a + 21 a - 512$

$b = - 7 a + 192$

$1024 = 16 a + 2 (- 7 a + 192) + 64$

Add $- 9 a$ and $21 a$ .

$c = 12 a - 512$

$b = - 7 a + 192$

$1024 = 16 a + 2 (- 7 a + 192) + 64$

$c = 12 a - 512$

$b = - 7 a + 192$

$1024 = 16 a + 2 (- 7 a + 192) + 64$

$c = 12 a - 512$

$b = - 7 a + 192$

$1024 = 16 a + 2 (- 7 a + 192) + 64$

Simplify $16 a + 2 (- 7 a + 192) + 64$ .

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Simplify each term.

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Apply the distributive property.

$c = 12 a - 512$

$b = - 7 a + 192$

$1024 = 16 a + 2 (- 7 a) + 2 \cdot 192 + 64$

Multiply $- 7$ by $2$ .

$c = 12 a - 512$

$b = - 7 a + 192$

$1024 = 16 a - 14 a + 2 \cdot 192 + 64$

Multiply $2$ by $192$ .

$c = 12 a - 512$

$b = - 7 a + 192$

$1024 = 16 a - 14 a + 384 + 64$

$c = 12 a - 512$

$b = - 7 a + 192$

$1024 = 16 a - 14 a + 384 + 64$

Simplify by adding terms.

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Subtract $14 a$ from $16 a$ .

$c = 12 a - 512$

$b = - 7 a + 192$

$1024 = 2 a + 384 + 64$

Add $384$ and $64$ .

$c = 12 a - 512$

$b = - 7 a + 192$

$1024 = 2 a + 448$

$c = 12 a - 512$

$b = - 7 a + 192$

$1024 = 2 a + 448$

$c = 12 a - 512$

$b = - 7 a + 192$

$1024 = 2 a + 448$

$c = 12 a - 512$

$b = - 7 a + 192$

$1024 = 2 a + 448$

Solve for $a$ in the third equation.

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Rewrite the equation as $2 a + 448 = 1024$ .

$c = 12 a - 512$

$b = - 7 a + 192$

$2 a + 448 = 1024$

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

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Subtract $448$ from both sides of the equation.

$c = 12 a - 512$

$b = - 7 a + 192$

$2 a = 1024 - 448$

Subtract $448$ from $1024$ .

$c = 12 a - 512$

$b = - 7 a + 192$

$2 a = 576$

$c = 12 a - 512$

$b = - 7 a + 192$

$2 a = 576$

Divide each term by $2$ and simplify.

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Divide each term in $2 a = 576$ by $2$ .

$c = 12 a - 512$

$b = - 7 a + 192$

$\frac{2 a}{2} = \frac{576}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$c = 12 a - 512$

$b = - 7 a + 192$

$\frac{2 a}{2} = \frac{576}{2}$

Divide $a$ by $1$ .

$c = 12 a - 512$

$b = - 7 a + 192$

$a = \frac{576}{2}$

$c = 12 a - 512$

$b = - 7 a + 192$

$a = \frac{576}{2}$

Divide $576$ by $2$ .

$c = 12 a - 512$

$b = - 7 a + 192$

$a = 288$

$c = 12 a - 512$

$b = - 7 a + 192$

$a = 288$

$c = 12 a - 512$

$b = - 7 a + 192$

$a = 288$

Replace all occurrences of $a$ with $288$ in each equation.

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Replace all occurrences of $a$ in $c = 12 a - 512$ with $288$ .

$c = 12 (288) - 512$

$b = - 7 a + 192$

$a = 288$

Replace all occurrences of $a$ in $b = - 7 a + 192$ with $288$ .

$c = 12 (288) - 512$

$b = - 7 \cdot 288 + 192$

$a = 288$

$c = 12 (288) - 512$

$b = - 7 \cdot 288 + 192$

$a = 288$

Simplify.

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Simplify $12 (288) - 512$ .

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Multiply $12$ by $288$ .

$c = 3456 - 512$

$b = - 7 \cdot 288 + 192$

$a = 288$

Subtract $512$ from $3456$ .

$c = 2944$

$b = - 7 \cdot 288 + 192$

$a = 288$

$c = 2944$

$b = - 7 \cdot 288 + 192$

$a = 288$

Simplify $- 7 \cdot 288 + 192$ .

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Multiply $- 7$ by $288$ .

$c = 2944$

$b = - 2016 + 192$

$a = 288$

Add $- 2016$ and $192$ .

$c = 2944$

$b = - 1824$

$a = 288$

$c = 2944$

$b = - 1824$

$a = 288$

$c = 2944$

$b = - 1824$

$a = 288$

$c = 2944$

$b = - 1824$

$a = 288$

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

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Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 288$ , $b = - 1824$ , $c = 2944$ , and $x = 3$ .

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Simplify each term.

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

$y = 288 \cdot 9 + (- 1824) \cdot (3) + 2944$

Multiply $288$ by $9$ .

$y = 2592 + (- 1824) \cdot (3) + 2944$

Multiply $- 1824$ by $3$ .

$y = 2592 - 5472 + 2944$

Simplify by adding and subtracting.

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Subtract $5472$ from $2592$ .

$y = - 2880 + 2944$

Add $- 2880$ and $2944$ .

$y = 64$

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = 64$ and $f (x) = 64$ .

$64 = 64$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 288$ , $b = - 1824$ , $c = 2944$ , and $x = 4$ .

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Simplify each term.

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

$y = 288 \cdot 16 + (- 1824) \cdot (4) + 2944$

Multiply $288$ by $16$ .

$y = 4608 + (- 1824) \cdot (4) + 2944$

Multiply $- 1824$ by $4$ .

$y = 4608 - 7296 + 2944$

Simplify by adding and subtracting.

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Subtract $7296$ from $4608$ .

$y = - 2688 + 2944$

Add $- 2688$ and $2944$ .

$y = 256$

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 4$ . This check passes since $y = 256$ and $f (x) = 256$ .

$256 = 256$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 288$ , $b = - 1824$ , $c = 2944$ , and $x = 5$ .

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Simplify each term.

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Raise $5$ to the power of $2$ .

$y = 288 \cdot 25 + (- 1824) \cdot (5) + 2944$

Multiply $288$ by $25$ .

$y = 7200 + (- 1824) \cdot (5) + 2944$

Multiply $- 1824$ by $5$ .

$y = 7200 - 9120 + 2944$

Simplify by adding and subtracting.

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Subtract $9120$ from $7200$ .

$y = - 1920 + 2944$

Add $- 1920$ and $2944$ .

$y = 1024$

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 5$ . This check passes since $y = 1024$ and $f (x) = 1024$ .

$1024 = 1024$

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

The function is quadratic

Since all $y = f (x)$ , the function is quadratic and follows the form $y = 288 x^{2} - 1824 x + 2944$ .

$y = 288 x^{2} - 1824 x + 2944$

Find the Function Rule table[[x,f(x)],[3,64],[4,256],[5,1024]]

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