Find the Function Rule table[[x,y],[-6,252],[-5,175],[-4,112]]

$\begin{array}{c} x & y \\ - 6 & 252 \\ - 5 & 175 \\ - 4 & 112 \end{array}$

Step 1

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 $y = a x + b$ .

$252 = a (- 6) + b 175 = a (- 5) + b 112 = a (- 4) + b$

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

Solve for $b$ in $252 = a (- 6) + b$ .

Rewrite the equation as $a (- 6) + b = 252$ .

$a (- 6) + b = 252$

$175 = a (- 5) + b$

$112 = a (- 4) + b$

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

$- 6 a + b = 252$

$175 = a (- 5) + b$

$112 = a (- 4) + b$

Add $6 a$ to both sides of the equation.

$b = 252 + 6 a$

$175 = a (- 5) + b$

$112 = a (- 4) + b$

$b = 252 + 6 a$

$175 = a (- 5) + b$

$112 = a (- 4) + b$

Replace all occurrences of $b$ with $252 + 6 a$ in each equation.

Replace all occurrences of $b$ in $175 = a (- 5) + b$ with $252 + 6 a$ .

$175 = a (- 5) + 252 + 6 a$

$b = 252 + 6 a$

$112 = a (- 4) + b$

Simplify $175 = a (- 5) + 252 + 6 a$ .

Simplify the left side.

Remove parentheses.

$175 = a (- 5) + 252 + 6 a$

$b = 252 + 6 a$

$112 = a (- 4) + b$

$175 = a (- 5) + 252 + 6 a$

$b = 252 + 6 a$

$112 = a (- 4) + b$

Simplify the right side.

Simplify $a (- 5) + 252 + 6 a$ .

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

$175 = - 5 a + 252 + 6 a$

$b = 252 + 6 a$

$112 = a (- 4) + b$

Add $- 5 a$ and $6 a$ .

$175 = a + 252$

$b = 252 + 6 a$

$112 = a (- 4) + b$

$175 = a + 252$

$b = 252 + 6 a$

$112 = a (- 4) + b$

$175 = a + 252$

$b = 252 + 6 a$

$112 = a (- 4) + b$

$175 = a + 252$

$b = 252 + 6 a$

$112 = a (- 4) + b$

Replace all occurrences of $b$ in $112 = a (- 4) + b$ with $252 + 6 a$ .

$112 = a (- 4) + 252 + 6 a$

$175 = a + 252$

$b = 252 + 6 a$

Simplify $112 = a (- 4) + 252 + 6 a$ .

Simplify the left side.

Remove parentheses.

$112 = a (- 4) + 252 + 6 a$

$175 = a + 252$

$b = 252 + 6 a$

$112 = a (- 4) + 252 + 6 a$

$175 = a + 252$

$b = 252 + 6 a$

Simplify the right side.

Simplify $a (- 4) + 252 + 6 a$ .

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

$112 = - 4 a + 252 + 6 a$

$175 = a + 252$

$b = 252 + 6 a$

Add $- 4 a$ and $6 a$ .

$112 = 2 a + 252$

$175 = a + 252$

$b = 252 + 6 a$

$112 = 2 a + 252$

$175 = a + 252$

$b = 252 + 6 a$

$112 = 2 a + 252$

$175 = a + 252$

$b = 252 + 6 a$

$112 = 2 a + 252$

$175 = a + 252$

$b = 252 + 6 a$

$112 = 2 a + 252$

$175 = a + 252$

$b = 252 + 6 a$

Solve for $a$ in $112 = 2 a + 252$ .

Rewrite the equation as $2 a + 252 = 112$ .

$2 a + 252 = 112$

$175 = a + 252$

$b = 252 + 6 a$

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

Subtract $252$ from both sides of the equation.

$2 a = 112 - 252$

$175 = a + 252$

$b = 252 + 6 a$

Subtract $252$ from $112$ .

$2 a = - 140$

$175 = a + 252$

$b = 252 + 6 a$

$2 a = - 140$

$175 = a + 252$

$b = 252 + 6 a$

Divide each term in $2 a = - 140$ by $2$ and simplify.

Divide each term in $2 a = - 140$ by $2$ .

$\frac{2 a}{2} = \frac{- 140}{2}$

$175 = a + 252$

$b = 252 + 6 a$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{- 140}{2}$

$175 = a + 252$

$b = 252 + 6 a$

Divide $a$ by $1$ .

$a = \frac{- 140}{2}$

$175 = a + 252$

$b = 252 + 6 a$

$a = \frac{- 140}{2}$

$175 = a + 252$

$b = 252 + 6 a$

$a = \frac{- 140}{2}$

$175 = a + 252$

$b = 252 + 6 a$

Simplify the right side.

Divide $- 140$ by $2$ .

$a = - 70$

$175 = a + 252$

$b = 252 + 6 a$

$a = - 70$

$175 = a + 252$

$b = 252 + 6 a$

$a = - 70$

$175 = a + 252$

$b = 252 + 6 a$

$a = - 70$

$175 = a + 252$

$b = 252 + 6 a$

Replace all occurrences of $a$ with $- 70$ in each equation.

Replace all occurrences of $a$ in $175 = a + 252$ with $- 70$ .

$175 = (- 70) + 252$

$a = - 70$

$b = 252 + 6 a$

Simplify the right side.

Add $- 70$ and $252$ .

$175 = 182$

$a = - 70$

$b = 252 + 6 a$

$175 = 182$

$a = - 70$

$b = 252 + 6 a$

Replace all occurrences of $a$ in $b = 252 + 6 a$ with $- 70$ .

$b = 252 + 6 (- 70)$

$175 = 182$

$a = - 70$

Simplify the right side.

Simplify $252 + 6 (- 70)$ .

Multiply $6$ by $- 70$ .

$b = 252 - 420$

$175 = 182$

$a = - 70$

Subtract $420$ from $252$ .

$b = - 168$

$175 = 182$

$a = - 70$

$b = - 168$

$175 = 182$

$a = - 70$

$b = - 168$

$175 = 182$

$a = - 70$

$b = - 168$

$175 = 182$

$a = - 70$

Since $175 = 182$ 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

Step 2

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 $c$ in $252 = a {(- 6)}^{2} + b (- 6) + c$ .

Rewrite the equation as $a {(- 6)}^{2} + b (- 6) + c = 252$ .

$a {(- 6)}^{2} + b (- 6) + c = 252$

$175 = a {(- 5)}^{2} + b (- 5) + c$

$112 = a {(- 4)}^{2} + b (- 4) + c$

Simplify each term.

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

$a \cdot 36 + b (- 6) + c = 252$

$175 = a {(- 5)}^{2} + b (- 5) + c$

$112 = a {(- 4)}^{2} + b (- 4) + c$

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

$36 \cdot a + b (- 6) + c = 252$

$175 = a {(- 5)}^{2} + b (- 5) + c$

$112 = a {(- 4)}^{2} + b (- 4) + c$

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

$36 a - 6 b + c = 252$

$175 = a {(- 5)}^{2} + b (- 5) + c$

$112 = a {(- 4)}^{2} + b (- 4) + c$

$36 a - 6 b + c = 252$

$175 = a {(- 5)}^{2} + b (- 5) + c$

$112 = a {(- 4)}^{2} + b (- 4) + c$

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

Subtract $36 a$ from both sides of the equation.

$- 6 b + c = 252 - 36 a$

$175 = a {(- 5)}^{2} + b (- 5) + c$

$112 = a {(- 4)}^{2} + b (- 4) + c$

Add $6 b$ to both sides of the equation.

$c = 252 - 36 a + 6 b$

$175 = a {(- 5)}^{2} + b (- 5) + c$

$112 = a {(- 4)}^{2} + b (- 4) + c$

$c = 252 - 36 a + 6 b$

$175 = a {(- 5)}^{2} + b (- 5) + c$

$112 = a {(- 4)}^{2} + b (- 4) + c$

$c = 252 - 36 a + 6 b$

$175 = a {(- 5)}^{2} + b (- 5) + c$

$112 = a {(- 4)}^{2} + b (- 4) + c$

Replace all occurrences of $c$ with $252 - 36 a + 6 b$ in each equation.

Replace all occurrences of $c$ in $175 = a {(- 5)}^{2} + b (- 5) + c$ with $252 - 36 a + 6 b$ .

$175 = a {(- 5)}^{2} + b (- 5) + 252 - 36 a + 6 b$

$c = 252 - 36 a + 6 b$

$112 = a {(- 4)}^{2} + b (- 4) + c$

Simplify $175 = a {(- 5)}^{2} + b (- 5) + 252 - 36 a + 6 b$ .

Simplify the left side.

Remove parentheses.

$175 = a {(- 5)}^{2} + b (- 5) + 252 - 36 a + 6 b$

$c = 252 - 36 a + 6 b$

$112 = a {(- 4)}^{2} + b (- 4) + c$

$175 = a {(- 5)}^{2} + b (- 5) + 252 - 36 a + 6 b$

$c = 252 - 36 a + 6 b$

$112 = a {(- 4)}^{2} + b (- 4) + c$

Simplify the right side.

Simplify $a {(- 5)}^{2} + b (- 5) + 252 - 36 a + 6 b$ .

Simplify each term.

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

$175 = a \cdot 25 + b (- 5) + 252 - 36 a + 6 b$

$c = 252 - 36 a + 6 b$

$112 = a {(- 4)}^{2} + b (- 4) + c$

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

$175 = 25 \cdot a + b (- 5) + 252 - 36 a + 6 b$

$c = 252 - 36 a + 6 b$

$112 = a {(- 4)}^{2} + b (- 4) + c$

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

$175 = 25 a - 5 b + 252 - 36 a + 6 b$

$c = 252 - 36 a + 6 b$

$112 = a {(- 4)}^{2} + b (- 4) + c$

$175 = 25 a - 5 b + 252 - 36 a + 6 b$

$c = 252 - 36 a + 6 b$

$112 = a {(- 4)}^{2} + b (- 4) + c$

Simplify by adding terms.

Subtract $36 a$ from $25 a$ .

$175 = - 11 a - 5 b + 252 + 6 b$

$c = 252 - 36 a + 6 b$

$112 = a {(- 4)}^{2} + b (- 4) + c$

Add $- 5 b$ and $6 b$ .

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

$112 = a {(- 4)}^{2} + b (- 4) + c$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

$112 = a {(- 4)}^{2} + b (- 4) + c$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

$112 = a {(- 4)}^{2} + b (- 4) + c$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

$112 = a {(- 4)}^{2} + b (- 4) + c$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

$112 = a {(- 4)}^{2} + b (- 4) + c$

Replace all occurrences of $c$ in $112 = a {(- 4)}^{2} + b (- 4) + c$ with $252 - 36 a + 6 b$ .

$112 = a {(- 4)}^{2} + b (- 4) + 252 - 36 a + 6 b$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

Simplify $112 = a {(- 4)}^{2} + b (- 4) + 252 - 36 a + 6 b$ .

Simplify the left side.

Remove parentheses.

$112 = a {(- 4)}^{2} + b (- 4) + 252 - 36 a + 6 b$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

$112 = a {(- 4)}^{2} + b (- 4) + 252 - 36 a + 6 b$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

Simplify the right side.

Simplify $a {(- 4)}^{2} + b (- 4) + 252 - 36 a + 6 b$ .

Simplify each term.

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

$112 = a \cdot 16 + b (- 4) + 252 - 36 a + 6 b$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

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

$112 = 16 \cdot a + b (- 4) + 252 - 36 a + 6 b$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

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

$112 = 16 a - 4 b + 252 - 36 a + 6 b$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

$112 = 16 a - 4 b + 252 - 36 a + 6 b$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

Simplify by adding terms.

Subtract $36 a$ from $16 a$ .

$112 = - 20 a - 4 b + 252 + 6 b$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

Add $- 4 b$ and $6 b$ .

$112 = - 20 a + 2 b + 252$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

$112 = - 20 a + 2 b + 252$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

$112 = - 20 a + 2 b + 252$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

$112 = - 20 a + 2 b + 252$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

$112 = - 20 a + 2 b + 252$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

$112 = - 20 a + 2 b + 252$

$175 = - 11 a + b + 252$

$c = 252 - 36 a + 6 b$

Solve for $b$ in $175 = - 11 a + b + 252$ .

Rewrite the equation as $- 11 a + b + 252 = 175$ .

$- 11 a + b + 252 = 175$

$112 = - 20 a + 2 b + 252$

$c = 252 - 36 a + 6 b$

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

Add $11 a$ to both sides of the equation.

$b + 252 = 175 + 11 a$

$112 = - 20 a + 2 b + 252$

$c = 252 - 36 a + 6 b$

Subtract $252$ from both sides of the equation.

$b = 175 + 11 a - 252$

$112 = - 20 a + 2 b + 252$

$c = 252 - 36 a + 6 b$

Subtract $252$ from $175$ .

$b = 11 a - 77$

$112 = - 20 a + 2 b + 252$

$c = 252 - 36 a + 6 b$

$b = 11 a - 77$

$112 = - 20 a + 2 b + 252$

$c = 252 - 36 a + 6 b$

$b = 11 a - 77$

$112 = - 20 a + 2 b + 252$

$c = 252 - 36 a + 6 b$

Replace all occurrences of $b$ with $11 a - 77$ in each equation.

Replace all occurrences of $b$ in $112 = - 20 a + 2 b + 252$ with $11 a - 77$ .

$112 = - 20 a + 2 (11 a - 77) + 252$

$b = 11 a - 77$

$c = 252 - 36 a + 6 b$

Simplify the right side.

Simplify $- 20 a + 2 (11 a - 77) + 252$ .

Simplify each term.

Apply the distributive property.

$112 = - 20 a + 2 (11 a) + 2 \cdot - 77 + 252$

$b = 11 a - 77$

$c = 252 - 36 a + 6 b$

Multiply $11$ by $2$ .

$112 = - 20 a + 22 a + 2 \cdot - 77 + 252$

$b = 11 a - 77$

$c = 252 - 36 a + 6 b$

Multiply $2$ by $- 77$ .

$112 = - 20 a + 22 a - 154 + 252$

$b = 11 a - 77$

$c = 252 - 36 a + 6 b$

$112 = - 20 a + 22 a - 154 + 252$

$b = 11 a - 77$

$c = 252 - 36 a + 6 b$

Simplify by adding terms.

Add $- 20 a$ and $22 a$ .

$112 = 2 a - 154 + 252$

$b = 11 a - 77$

$c = 252 - 36 a + 6 b$

Add $- 154$ and $252$ .

$112 = 2 a + 98$

$b = 11 a - 77$

$c = 252 - 36 a + 6 b$

$112 = 2 a + 98$

$b = 11 a - 77$

$c = 252 - 36 a + 6 b$

$112 = 2 a + 98$

$b = 11 a - 77$

$c = 252 - 36 a + 6 b$

$112 = 2 a + 98$

$b = 11 a - 77$

$c = 252 - 36 a + 6 b$

Replace all occurrences of $b$ in $c = 252 - 36 a + 6 b$ with $11 a - 77$ .

$c = 252 - 36 a + 6 (11 a - 77)$

$112 = 2 a + 98$

$b = 11 a - 77$

Simplify the right side.

Simplify $252 - 36 a + 6 (11 a - 77)$ .

Simplify each term.

Apply the distributive property.

$c = 252 - 36 a + 6 (11 a) + 6 \cdot - 77$

$112 = 2 a + 98$

$b = 11 a - 77$

Multiply $11$ by $6$ .

$c = 252 - 36 a + 66 a + 6 \cdot - 77$

$112 = 2 a + 98$

$b = 11 a - 77$

Multiply $6$ by $- 77$ .

$c = 252 - 36 a + 66 a - 462$

$112 = 2 a + 98$

$b = 11 a - 77$

$c = 252 - 36 a + 66 a - 462$

$112 = 2 a + 98$

$b = 11 a - 77$

Simplify by adding terms.

Subtract $462$ from $252$ .

$c = - 36 a + 66 a - 210$

$112 = 2 a + 98$

$b = 11 a - 77$

Add $- 36 a$ and $66 a$ .

$c = 30 a - 210$

$112 = 2 a + 98$

$b = 11 a - 77$

$c = 30 a - 210$

$112 = 2 a + 98$

$b = 11 a - 77$

$c = 30 a - 210$

$112 = 2 a + 98$

$b = 11 a - 77$

$c = 30 a - 210$

$112 = 2 a + 98$

$b = 11 a - 77$

$c = 30 a - 210$

$112 = 2 a + 98$

$b = 11 a - 77$

Solve for $a$ in $112 = 2 a + 98$ .

Rewrite the equation as $2 a + 98 = 112$ .

$2 a + 98 = 112$

$c = 30 a - 210$

$b = 11 a - 77$

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

Subtract $98$ from both sides of the equation.

$2 a = 112 - 98$

$c = 30 a - 210$

$b = 11 a - 77$

Subtract $98$ from $112$ .

$2 a = 14$

$c = 30 a - 210$

$b = 11 a - 77$

$2 a = 14$

$c = 30 a - 210$

$b = 11 a - 77$

Divide each term in $2 a = 14$ by $2$ and simplify.

Divide each term in $2 a = 14$ by $2$ .

$\frac{2 a}{2} = \frac{14}{2}$

$c = 30 a - 210$

$b = 11 a - 77$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{14}{2}$

$c = 30 a - 210$

$b = 11 a - 77$

Divide $a$ by $1$ .

$a = \frac{14}{2}$

$c = 30 a - 210$

$b = 11 a - 77$

$a = \frac{14}{2}$

$c = 30 a - 210$

$b = 11 a - 77$

$a = \frac{14}{2}$

$c = 30 a - 210$

$b = 11 a - 77$

Simplify the right side.

Divide $14$ by $2$ .

$a = 7$

$c = 30 a - 210$

$b = 11 a - 77$

$a = 7$

$c = 30 a - 210$

$b = 11 a - 77$

$a = 7$

$c = 30 a - 210$

$b = 11 a - 77$

$a = 7$

$c = 30 a - 210$

$b = 11 a - 77$

Replace all occurrences of $a$ with $7$ in each equation.

Replace all occurrences of $a$ in $c = 30 a - 210$ with $7$ .

$c = 30 (7) - 210$

$a = 7$

$b = 11 a - 77$

Simplify the right side.

Simplify $30 (7) - 210$ .

Multiply $30$ by $7$ .

$c = 210 - 210$

$a = 7$

$b = 11 a - 77$

Subtract $210$ from $210$ .

$c = 0$

$a = 7$

$b = 11 a - 77$

$c = 0$

$a = 7$

$b = 11 a - 77$

$c = 0$

$a = 7$

$b = 11 a - 77$

Replace all occurrences of $a$ in $b = 11 a - 77$ with $7$ .

$b = 11 (7) - 77$

$c = 0$

$a = 7$

Simplify the right side.

Simplify $11 (7) - 77$ .

Multiply $11$ by $7$ .

$b = 77 - 77$

$c = 0$

$a = 7$

Subtract $77$ from $77$ .

$b = 0$

$c = 0$

$a = 7$

$b = 0$

$c = 0$

$a = 7$

$b = 0$

$c = 0$

$a = 7$

$b = 0$

$c = 0$

$a = 7$

List all of the solutions.

$b = 0, c = 0, a = 7$

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 $a = 7$ , $b = 0$ , $c = 0$ , and $x = - 6$ .

Simplify each term.

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

$y = 7 \cdot 36 + (0) \cdot (- 6) + 0$

Multiply $7$ by $36$ .

$y = 252 + (0) \cdot (- 6) + 0$

Multiply $0$ by $- 6$ .

$y = 252 + 0 + 0$

Simplify by adding numbers.

Add $252$ and $0$ .

$y = 252 + 0$

Add $252$ and $0$ .

$y = 252$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = - 6$ . This check passes since $y = 252$ and $y = 252$ .

$252 = 252$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 7$ , $b = 0$ , $c = 0$ , and $x = - 5$ .

Simplify each term.

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

$y = 7 \cdot 25 + (0) \cdot (- 5) + 0$

Multiply $7$ by $25$ .

$y = 175 + (0) \cdot (- 5) + 0$

Multiply $0$ by $- 5$ .

$y = 175 + 0 + 0$

Simplify by adding numbers.

Add $175$ and $0$ .

$y = 175 + 0$

Add $175$ and $0$ .

$y = 175$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = - 5$ . This check passes since $y = 175$ and $y = 175$ .

$175 = 175$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 7$ , $b = 0$ , $c = 0$ , and $x = - 4$ .

Simplify each term.

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

$y = 7 \cdot 16 + (0) \cdot (- 4) + 0$

Multiply $7$ by $16$ .

$y = 112 + (0) \cdot (- 4) + 0$

Multiply $0$ by $- 4$ .

$y = 112 + 0 + 0$

Simplify by adding numbers.

Add $112$ and $0$ .

$y = 112 + 0$

Add $112$ and $0$ .

$y = 112$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = - 4$ . This check passes since $y = 112$ and $y = 112$ .

$112 = 112$

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

The function is quadratic

Step 3

Since all $y = y$ , the function is quadratic and follows the form $y = 7 x^{2}$ .

$y = 7 x^{2}$

Find the Function Rule table[[x,y],[-6,252],[-5,175],[-4,112]]

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