Find the Function Rule table[[x,y],[-5,-36],[-4,-25],[-3,-16]]

$\begin{array}{c} x & y \\ - 5 & - 36 \\ - 4 & - 25 \\ - 3 & - 16 \end{array}$

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 $y = a x + b$ .

$- 36 = a (- 5) + b - 25 = a (- 4) + b - 16 = a (- 3) + b$

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

Solve for $b$ in $- 36 = a (- 5) + b$ .

Rewrite the equation as $a (- 5) + b = - 36$ .

$a (- 5) + b = - 36$

$- 25 = a (- 4) + b$

$- 16 = a (- 3) + b$

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

$- 5 a + b = - 36$

$- 25 = a (- 4) + b$

$- 16 = a (- 3) + b$

Add $5 a$ to both sides of the equation.

$b = - 36 + 5 a$

$- 25 = a (- 4) + b$

$- 16 = a (- 3) + b$

$b = - 36 + 5 a$

$- 25 = a (- 4) + b$

$- 16 = a (- 3) + b$

Replace all occurrences of $b$ with $- 36 + 5 a$ in each equation.

Replace all occurrences of $b$ in $- 25 = a (- 4) + b$ with $- 36 + 5 a$ .

$- 25 = a (- 4) - 36 + 5 a$

$b = - 36 + 5 a$

$- 16 = a (- 3) + b$

Simplify $- 25 = a (- 4) - 36 + 5 a$ .

Simplify the left side.

Remove parentheses.

$- 25 = a (- 4) - 36 + 5 a$

$b = - 36 + 5 a$

$- 16 = a (- 3) + b$

$- 25 = a (- 4) - 36 + 5 a$

$b = - 36 + 5 a$

$- 16 = a (- 3) + b$

Simplify the right side.

Simplify $a (- 4) - 36 + 5 a$ .

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

$- 25 = - 4 a - 36 + 5 a$

$b = - 36 + 5 a$

$- 16 = a (- 3) + b$

Add $- 4 a$ and $5 a$ .

$- 25 = a - 36$

$b = - 36 + 5 a$

$- 16 = a (- 3) + b$

$- 25 = a - 36$

$b = - 36 + 5 a$

$- 16 = a (- 3) + b$

$- 25 = a - 36$

$b = - 36 + 5 a$

$- 16 = a (- 3) + b$

$- 25 = a - 36$

$b = - 36 + 5 a$

$- 16 = a (- 3) + b$

Replace all occurrences of $b$ in $- 16 = a (- 3) + b$ with $- 36 + 5 a$ .

$- 16 = a (- 3) - 36 + 5 a$

$- 25 = a - 36$

$b = - 36 + 5 a$

Simplify $- 16 = a (- 3) - 36 + 5 a$ .

Simplify the left side.

Remove parentheses.

$- 16 = a (- 3) - 36 + 5 a$

$- 25 = a - 36$

$b = - 36 + 5 a$

$- 16 = a (- 3) - 36 + 5 a$

$- 25 = a - 36$

$b = - 36 + 5 a$

Simplify the right side.

Simplify $a (- 3) - 36 + 5 a$ .

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

$- 16 = - 3 a - 36 + 5 a$

$- 25 = a - 36$

$b = - 36 + 5 a$

Add $- 3 a$ and $5 a$ .

$- 16 = 2 a - 36$

$- 25 = a - 36$

$b = - 36 + 5 a$

$- 16 = 2 a - 36$

$- 25 = a - 36$

$b = - 36 + 5 a$

$- 16 = 2 a - 36$

$- 25 = a - 36$

$b = - 36 + 5 a$

$- 16 = 2 a - 36$

$- 25 = a - 36$

$b = - 36 + 5 a$

$- 16 = 2 a - 36$

$- 25 = a - 36$

$b = - 36 + 5 a$

Solve for $a$ in $- 16 = 2 a - 36$ .

Rewrite the equation as $2 a - 36 = - 16$ .

$2 a - 36 = - 16$

$- 25 = a - 36$

$b = - 36 + 5 a$

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

Add $36$ to both sides of the equation.

$2 a = - 16 + 36$

$- 25 = a - 36$

$b = - 36 + 5 a$

Add $- 16$ and $36$ .

$2 a = 20$

$- 25 = a - 36$

$b = - 36 + 5 a$

$2 a = 20$

$- 25 = a - 36$

$b = - 36 + 5 a$

Divide each term in $2 a = 20$ by $2$ and simplify.

Divide each term in $2 a = 20$ by $2$ .

$\frac{2 a}{2} = \frac{20}{2}$

$- 25 = a - 36$

$b = - 36 + 5 a$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{20}{2}$

$- 25 = a - 36$

$b = - 36 + 5 a$

Divide $a$ by $1$ .

$a = \frac{20}{2}$

$- 25 = a - 36$

$b = - 36 + 5 a$

$a = \frac{20}{2}$

$- 25 = a - 36$

$b = - 36 + 5 a$

$a = \frac{20}{2}$

$- 25 = a - 36$

$b = - 36 + 5 a$

Simplify the right side.

Divide $20$ by $2$ .

$a = 10$

$- 25 = a - 36$

$b = - 36 + 5 a$

$a = 10$

$- 25 = a - 36$

$b = - 36 + 5 a$

$a = 10$

$- 25 = a - 36$

$b = - 36 + 5 a$

$a = 10$

$- 25 = a - 36$

$b = - 36 + 5 a$

Replace all occurrences of $a$ with $10$ in each equation.

Replace all occurrences of $a$ in $- 25 = a - 36$ with $10$ .

$- 25 = (10) - 36$

$a = 10$

$b = - 36 + 5 a$

Simplify the right side.

Subtract $36$ from $10$ .

$- 25 = - 26$

$a = 10$

$b = - 36 + 5 a$

$- 25 = - 26$

$a = 10$

$b = - 36 + 5 a$

Replace all occurrences of $a$ in $b = - 36 + 5 a$ with $10$ .

$b = - 36 + 5 (10)$

$- 25 = - 26$

$a = 10$

Simplify the right side.

Simplify $- 36 + 5 (10)$ .

Multiply $5$ by $10$ .

$b = - 36 + 50$

$- 25 = - 26$

$a = 10$

Add $- 36$ and $50$ .

$b = 14$

$- 25 = - 26$

$a = 10$

$b = 14$

$- 25 = - 26$

$a = 10$

$b = 14$

$- 25 = - 26$

$a = 10$

$b = 14$

$- 25 = - 26$

$a = 10$

Since $- 25 = - 26$ 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 $- 36 = a {(- 5)}^{2} + b (- 5) + c$ .

Rewrite the equation as $a {(- 5)}^{2} + b (- 5) + c = - 36$ .

$a {(- 5)}^{2} + b (- 5) + c = - 36$

$- 25 = a {(- 4)}^{2} + b (- 4) + c$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

Simplify each term.

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

$a \cdot 25 + b (- 5) + c = - 36$

$- 25 = a {(- 4)}^{2} + b (- 4) + c$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

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

$25 \cdot a + b (- 5) + c = - 36$

$- 25 = a {(- 4)}^{2} + b (- 4) + c$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

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

$25 a - 5 b + c = - 36$

$- 25 = a {(- 4)}^{2} + b (- 4) + c$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

$25 a - 5 b + c = - 36$

$- 25 = a {(- 4)}^{2} + b (- 4) + c$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

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

Subtract $25 a$ from both sides of the equation.

$- 5 b + c = - 36 - 25 a$

$- 25 = a {(- 4)}^{2} + b (- 4) + c$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

Add $5 b$ to both sides of the equation.

$c = - 36 - 25 a + 5 b$

$- 25 = a {(- 4)}^{2} + b (- 4) + c$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

$c = - 36 - 25 a + 5 b$

$- 25 = a {(- 4)}^{2} + b (- 4) + c$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

$c = - 36 - 25 a + 5 b$

$- 25 = a {(- 4)}^{2} + b (- 4) + c$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

Replace all occurrences of $c$ with $- 36 - 25 a + 5 b$ in each equation.

Replace all occurrences of $c$ in $- 25 = a {(- 4)}^{2} + b (- 4) + c$ with $- 36 - 25 a + 5 b$ .

$- 25 = a {(- 4)}^{2} + b (- 4) - 36 - 25 a + 5 b$

$c = - 36 - 25 a + 5 b$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

Simplify $- 25 = a {(- 4)}^{2} + b (- 4) - 36 - 25 a + 5 b$ .

Simplify the left side.

Remove parentheses.

$- 25 = a {(- 4)}^{2} + b (- 4) - 36 - 25 a + 5 b$

$c = - 36 - 25 a + 5 b$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

$- 25 = a {(- 4)}^{2} + b (- 4) - 36 - 25 a + 5 b$

$c = - 36 - 25 a + 5 b$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

Simplify the right side.

Simplify $a {(- 4)}^{2} + b (- 4) - 36 - 25 a + 5 b$ .

Simplify each term.

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

$- 25 = a \cdot 16 + b (- 4) - 36 - 25 a + 5 b$

$c = - 36 - 25 a + 5 b$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

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

$- 25 = 16 \cdot a + b (- 4) - 36 - 25 a + 5 b$

$c = - 36 - 25 a + 5 b$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

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

$- 25 = 16 a - 4 b - 36 - 25 a + 5 b$

$c = - 36 - 25 a + 5 b$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

$- 25 = 16 a - 4 b - 36 - 25 a + 5 b$

$c = - 36 - 25 a + 5 b$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

Simplify by adding terms.

Subtract $25 a$ from $16 a$ .

$- 25 = - 9 a - 4 b - 36 + 5 b$

$c = - 36 - 25 a + 5 b$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

Add $- 4 b$ and $5 b$ .

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

$- 16 = a {(- 3)}^{2} + b (- 3) + c$

Replace all occurrences of $c$ in $- 16 = a {(- 3)}^{2} + b (- 3) + c$ with $- 36 - 25 a + 5 b$ .

$- 16 = a {(- 3)}^{2} + b (- 3) - 36 - 25 a + 5 b$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

Simplify $- 16 = a {(- 3)}^{2} + b (- 3) - 36 - 25 a + 5 b$ .

Simplify the left side.

Remove parentheses.

$- 16 = a {(- 3)}^{2} + b (- 3) - 36 - 25 a + 5 b$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

$- 16 = a {(- 3)}^{2} + b (- 3) - 36 - 25 a + 5 b$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

Simplify the right side.

Simplify $a {(- 3)}^{2} + b (- 3) - 36 - 25 a + 5 b$ .

Simplify each term.

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

$- 16 = a \cdot 9 + b (- 3) - 36 - 25 a + 5 b$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

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

$- 16 = 9 \cdot a + b (- 3) - 36 - 25 a + 5 b$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

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

$- 16 = 9 a - 3 b - 36 - 25 a + 5 b$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

$- 16 = 9 a - 3 b - 36 - 25 a + 5 b$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

Simplify by adding terms.

Subtract $25 a$ from $9 a$ .

$- 16 = - 16 a - 3 b - 36 + 5 b$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

Add $- 3 b$ and $5 b$ .

$- 16 = - 16 a + 2 b - 36$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

$- 16 = - 16 a + 2 b - 36$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

$- 16 = - 16 a + 2 b - 36$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

$- 16 = - 16 a + 2 b - 36$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

$- 16 = - 16 a + 2 b - 36$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

$- 16 = - 16 a + 2 b - 36$

$- 25 = - 9 a + b - 36$

$c = - 36 - 25 a + 5 b$

Solve for $b$ in $- 25 = - 9 a + b - 36$ .

Rewrite the equation as $- 9 a + b - 36 = - 25$ .

$- 9 a + b - 36 = - 25$

$- 16 = - 16 a + 2 b - 36$

$c = - 36 - 25 a + 5 b$

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

Add $9 a$ to both sides of the equation.

$b - 36 = - 25 + 9 a$

$- 16 = - 16 a + 2 b - 36$

$c = - 36 - 25 a + 5 b$

Add $36$ to both sides of the equation.

$b = - 25 + 9 a + 36$

$- 16 = - 16 a + 2 b - 36$

$c = - 36 - 25 a + 5 b$

Add $- 25$ and $36$ .

$b = 9 a + 11$

$- 16 = - 16 a + 2 b - 36$

$c = - 36 - 25 a + 5 b$

$b = 9 a + 11$

$- 16 = - 16 a + 2 b - 36$

$c = - 36 - 25 a + 5 b$

$b = 9 a + 11$

$- 16 = - 16 a + 2 b - 36$

$c = - 36 - 25 a + 5 b$

Replace all occurrences of $b$ with $9 a + 11$ in each equation.

Replace all occurrences of $b$ in $- 16 = - 16 a + 2 b - 36$ with $9 a + 11$ .

$- 16 = - 16 a + 2 (9 a + 11) - 36$

$b = 9 a + 11$

$c = - 36 - 25 a + 5 b$

Simplify the right side.

Simplify $- 16 a + 2 (9 a + 11) - 36$ .

Simplify each term.

Apply the distributive property.

$- 16 = - 16 a + 2 (9 a) + 2 \cdot 11 - 36$

$b = 9 a + 11$

$c = - 36 - 25 a + 5 b$

Multiply $9$ by $2$ .

$- 16 = - 16 a + 18 a + 2 \cdot 11 - 36$

$b = 9 a + 11$

$c = - 36 - 25 a + 5 b$

Multiply $2$ by $11$ .

$- 16 = - 16 a + 18 a + 22 - 36$

$b = 9 a + 11$

$c = - 36 - 25 a + 5 b$

$- 16 = - 16 a + 18 a + 22 - 36$

$b = 9 a + 11$

$c = - 36 - 25 a + 5 b$

Simplify by adding terms.

Add $- 16 a$ and $18 a$ .

$- 16 = 2 a + 22 - 36$

$b = 9 a + 11$

$c = - 36 - 25 a + 5 b$

Subtract $36$ from $22$ .

$- 16 = 2 a - 14$

$b = 9 a + 11$

$c = - 36 - 25 a + 5 b$

$- 16 = 2 a - 14$

$b = 9 a + 11$

$c = - 36 - 25 a + 5 b$

$- 16 = 2 a - 14$

$b = 9 a + 11$

$c = - 36 - 25 a + 5 b$

$- 16 = 2 a - 14$

$b = 9 a + 11$

$c = - 36 - 25 a + 5 b$

Replace all occurrences of $b$ in $c = - 36 - 25 a + 5 b$ with $9 a + 11$ .

$c = - 36 - 25 a + 5 (9 a + 11)$

$- 16 = 2 a - 14$

$b = 9 a + 11$

Simplify the right side.

Simplify $- 36 - 25 a + 5 (9 a + 11)$ .

Simplify each term.

Apply the distributive property.

$c = - 36 - 25 a + 5 (9 a) + 5 \cdot 11$

$- 16 = 2 a - 14$

$b = 9 a + 11$

Multiply $9$ by $5$ .

$c = - 36 - 25 a + 45 a + 5 \cdot 11$

$- 16 = 2 a - 14$

$b = 9 a + 11$

Multiply $5$ by $11$ .

$c = - 36 - 25 a + 45 a + 55$

$- 16 = 2 a - 14$

$b = 9 a + 11$

$c = - 36 - 25 a + 45 a + 55$

$- 16 = 2 a - 14$

$b = 9 a + 11$

Simplify by adding terms.

Add $- 36$ and $55$ .

$c = - 25 a + 45 a + 19$

$- 16 = 2 a - 14$

$b = 9 a + 11$

Add $- 25 a$ and $45 a$ .

$c = 20 a + 19$

$- 16 = 2 a - 14$

$b = 9 a + 11$

$c = 20 a + 19$

$- 16 = 2 a - 14$

$b = 9 a + 11$

$c = 20 a + 19$

$- 16 = 2 a - 14$

$b = 9 a + 11$

$c = 20 a + 19$

$- 16 = 2 a - 14$

$b = 9 a + 11$

$c = 20 a + 19$

$- 16 = 2 a - 14$

$b = 9 a + 11$

Solve for $a$ in $- 16 = 2 a - 14$ .

Rewrite the equation as $2 a - 14 = - 16$ .

$2 a - 14 = - 16$

$c = 20 a + 19$

$b = 9 a + 11$

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

Add $14$ to both sides of the equation.

$2 a = - 16 + 14$

$c = 20 a + 19$

$b = 9 a + 11$

Add $- 16$ and $14$ .

$2 a = - 2$

$c = 20 a + 19$

$b = 9 a + 11$

$2 a = - 2$

$c = 20 a + 19$

$b = 9 a + 11$

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

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

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

$c = 20 a + 19$

$b = 9 a + 11$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

$c = 20 a + 19$

$b = 9 a + 11$

Divide $a$ by $1$ .

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

$c = 20 a + 19$

$b = 9 a + 11$

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

$c = 20 a + 19$

$b = 9 a + 11$

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

$c = 20 a + 19$

$b = 9 a + 11$

Simplify the right side.

Divide $- 2$ by $2$ .

$a = - 1$

$c = 20 a + 19$

$b = 9 a + 11$

$a = - 1$

$c = 20 a + 19$

$b = 9 a + 11$

$a = - 1$

$c = 20 a + 19$

$b = 9 a + 11$

$a = - 1$

$c = 20 a + 19$

$b = 9 a + 11$

Replace all occurrences of $a$ with $- 1$ in each equation.

Replace all occurrences of $a$ in $c = 20 a + 19$ with $- 1$ .

$c = 20 (- 1) + 19$

$a = - 1$

$b = 9 a + 11$

Simplify the right side.

Simplify $20 (- 1) + 19$ .

Multiply $20$ by $- 1$ .

$c = - 20 + 19$

$a = - 1$

$b = 9 a + 11$

Add $- 20$ and $19$ .

$c = - 1$

$a = - 1$

$b = 9 a + 11$

$c = - 1$

$a = - 1$

$b = 9 a + 11$

$c = - 1$

$a = - 1$

$b = 9 a + 11$

Replace all occurrences of $a$ in $b = 9 a + 11$ with $- 1$ .

$b = 9 (- 1) + 11$

$c = - 1$

$a = - 1$

Simplify the right side.

Simplify $9 (- 1) + 11$ .

Multiply $9$ by $- 1$ .

$b = - 9 + 11$

$c = - 1$

$a = - 1$

Add $- 9$ and $11$ .

$b = 2$

$c = - 1$

$a = - 1$

$b = 2$

$c = - 1$

$a = - 1$

$b = 2$

$c = - 1$

$a = - 1$

$b = 2$

$c = - 1$

$a = - 1$

List all of the solutions.

$b = 2, c = - 1, a = - 1$

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 $a = - 1$ , $b = 2$ , $c = - 1$ , and $x = - 5$ .

Simplify each term.

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

$y = - 1 \cdot 25 + (2) \cdot (- 5) - 1$

Multiply $- 1$ by $25$ .

$y = - 25 + (2) \cdot (- 5) - 1$

Multiply $2$ by $- 5$ .

$y = - 25 - 10 - 1$

Simplify by subtracting numbers.

Subtract $10$ from $- 25$ .

$y = - 35 - 1$

Subtract $1$ from $- 35$ .

$y = - 36$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = - 5$ . This check passes since $y = - 36$ and $y = - 36$ .

$- 36 = - 36$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = - 1$ , $b = 2$ , $c = - 1$ , and $x = - 4$ .

Simplify each term.

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

$y = - 1 \cdot 16 + (2) \cdot (- 4) - 1$

Multiply $- 1$ by $16$ .

$y = - 16 + (2) \cdot (- 4) - 1$

Multiply $2$ by $- 4$ .

$y = - 16 - 8 - 1$

Simplify by subtracting numbers.

Subtract $8$ from $- 16$ .

$y = - 24 - 1$

Subtract $1$ from $- 24$ .

$y = - 25$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = - 4$ . This check passes since $y = - 25$ and $y = - 25$ .

$- 25 = - 25$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = - 1$ , $b = 2$ , $c = - 1$ , and $x = - 3$ .

Simplify each term.

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

$y = - 1 \cdot 9 + (2) \cdot (- 3) - 1$

Multiply $- 1$ by $9$ .

$y = - 9 + (2) \cdot (- 3) - 1$

Multiply $2$ by $- 3$ .

$y = - 9 - 6 - 1$

Simplify by subtracting numbers.

Subtract $6$ from $- 9$ .

$y = - 15 - 1$

Subtract $1$ from $- 15$ .

$y = - 16$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = - 3$ . This check passes since $y = - 16$ and $y = - 16$ .

$- 16 = - 16$

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 $y = - x^{2} + 2 x - 1$ .

$y = - x^{2} + 2 x - 1$

Find the Function Rule table[[x,y],[-5,-36],[-4,-25],[-3,-16]]

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