Find the Function Rule table[[x,y],[2,9],[4,21],[6,41]]

$\begin{array}{c} x & y \\ 2 & 9 \\ 4 & 21 \\ 6 & 41 \end{array}$

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 $y = a x + b$ .

$9 = a (2) + b 21 = a (4) + b 41 = a (6) + b$

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

Solve for $b$ in $9 = a (2) + b$ .

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

$a (2) + b = 9$

$21 = a (4) + b$

$41 = a (6) + b$

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

$2 a + b = 9$

$21 = a (4) + b$

$41 = a (6) + b$

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

$b = 9 - 2 a$

$21 = a (4) + b$

$41 = a (6) + b$

$b = 9 - 2 a$

$21 = a (4) + b$

$41 = a (6) + b$

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

Replace all occurrences of $b$ in $21 = a (4) + b$ with $9 - 2 a$ .

$21 = a (4) + 9 - 2 a$

$b = 9 - 2 a$

$41 = a (6) + b$

Simplify $21 = a (4) + 9 - 2 a$ .

Simplify the left side.

Remove parentheses.

$21 = a (4) + 9 - 2 a$

$b = 9 - 2 a$

$41 = a (6) + b$

$21 = a (4) + 9 - 2 a$

$b = 9 - 2 a$

$41 = a (6) + b$

Simplify the right side.

Simplify $a (4) + 9 - 2 a$ .

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

$21 = 4 a + 9 - 2 a$

$b = 9 - 2 a$

$41 = a (6) + b$

Subtract $2 a$ from $4 a$ .

$21 = 2 a + 9$

$b = 9 - 2 a$

$41 = a (6) + b$

$21 = 2 a + 9$

$b = 9 - 2 a$

$41 = a (6) + b$

$21 = 2 a + 9$

$b = 9 - 2 a$

$41 = a (6) + b$

$21 = 2 a + 9$

$b = 9 - 2 a$

$41 = a (6) + b$

Replace all occurrences of $b$ in $41 = a (6) + b$ with $9 - 2 a$ .

$41 = a (6) + 9 - 2 a$

$21 = 2 a + 9$

$b = 9 - 2 a$

Simplify $41 = a (6) + 9 - 2 a$ .

Simplify the left side.

Remove parentheses.

$41 = a (6) + 9 - 2 a$

$21 = 2 a + 9$

$b = 9 - 2 a$

$41 = a (6) + 9 - 2 a$

$21 = 2 a + 9$

$b = 9 - 2 a$

Simplify the right side.

Simplify $a (6) + 9 - 2 a$ .

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

$41 = 6 a + 9 - 2 a$

$21 = 2 a + 9$

$b = 9 - 2 a$

Subtract $2 a$ from $6 a$ .

$41 = 4 a + 9$

$21 = 2 a + 9$

$b = 9 - 2 a$

$41 = 4 a + 9$

$21 = 2 a + 9$

$b = 9 - 2 a$

$41 = 4 a + 9$

$21 = 2 a + 9$

$b = 9 - 2 a$

$41 = 4 a + 9$

$21 = 2 a + 9$

$b = 9 - 2 a$

$41 = 4 a + 9$

$21 = 2 a + 9$

$b = 9 - 2 a$

Solve for $a$ in $41 = 4 a + 9$ .

Rewrite the equation as $4 a + 9 = 41$ .

$4 a + 9 = 41$

$21 = 2 a + 9$

$b = 9 - 2 a$

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

Subtract $9$ from both sides of the equation.

$4 a = 41 - 9$

$21 = 2 a + 9$

$b = 9 - 2 a$

Subtract $9$ from $41$ .

$4 a = 32$

$21 = 2 a + 9$

$b = 9 - 2 a$

$4 a = 32$

$21 = 2 a + 9$

$b = 9 - 2 a$

Divide each term in $4 a = 32$ by $4$ and simplify.

Divide each term in $4 a = 32$ by $4$ .

$\frac{4 a}{4} = \frac{32}{4}$

$21 = 2 a + 9$

$b = 9 - 2 a$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 a}{4} = \frac{32}{4}$

$21 = 2 a + 9$

$b = 9 - 2 a$

Divide $a$ by $1$ .

$a = \frac{32}{4}$

$21 = 2 a + 9$

$b = 9 - 2 a$

$a = \frac{32}{4}$

$21 = 2 a + 9$

$b = 9 - 2 a$

$a = \frac{32}{4}$

$21 = 2 a + 9$

$b = 9 - 2 a$

Simplify the right side.

Divide $32$ by $4$ .

$a = 8$

$21 = 2 a + 9$

$b = 9 - 2 a$

$a = 8$

$21 = 2 a + 9$

$b = 9 - 2 a$

$a = 8$

$21 = 2 a + 9$

$b = 9 - 2 a$

$a = 8$

$21 = 2 a + 9$

$b = 9 - 2 a$

Replace all occurrences of $a$ with $8$ in each equation.

Replace all occurrences of $a$ in $21 = 2 a + 9$ with $8$ .

$21 = 2 (8) + 9$

$a = 8$

$b = 9 - 2 a$

Simplify the right side.

Simplify $2 (8) + 9$ .

Multiply $2$ by $8$ .

$21 = 16 + 9$

$a = 8$

$b = 9 - 2 a$

Add $16$ and $9$ .

$21 = 25$

$a = 8$

$b = 9 - 2 a$

$21 = 25$

$a = 8$

$b = 9 - 2 a$

$21 = 25$

$a = 8$

$b = 9 - 2 a$

Replace all occurrences of $a$ in $b = 9 - 2 a$ with $8$ .

$b = 9 - 2 \cdot 8$

$21 = 25$

$a = 8$

Simplify the right side.

Simplify $9 - 2 \cdot 8$ .

Multiply $- 2$ by $8$ .

$b = 9 - 16$

$21 = 25$

$a = 8$

Subtract $16$ from $9$ .

$b = - 7$

$21 = 25$

$a = 8$

$b = - 7$

$21 = 25$

$a = 8$

$b = - 7$

$21 = 25$

$a = 8$

$b = - 7$

$21 = 25$

$a = 8$

Since $21 = 25$ 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 $9 = a \cdot 2^{2} + b (2) + c$ .

Rewrite the equation as $a \cdot 2^{2} + b (2) + c = 9$ .

$a \cdot 2^{2} + b (2) + c = 9$

$21 = a \cdot 4^{2} + b (4) + c$

$41 = a \cdot 6^{2} + b (6) + c$

Simplify each term.

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

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

$21 = a \cdot 4^{2} + b (4) + c$

$41 = a \cdot 6^{2} + b (6) + c$

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

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

$21 = a \cdot 4^{2} + b (4) + c$

$41 = a \cdot 6^{2} + b (6) + c$

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

$4 a + 2 b + c = 9$

$21 = a \cdot 4^{2} + b (4) + c$

$41 = a \cdot 6^{2} + b (6) + c$

$4 a + 2 b + c = 9$

$21 = a \cdot 4^{2} + b (4) + c$

$41 = a \cdot 6^{2} + b (6) + c$

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

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

$2 b + c = 9 - 4 a$

$21 = a \cdot 4^{2} + b (4) + c$

$41 = a \cdot 6^{2} + b (6) + c$

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

$c = 9 - 4 a - 2 b$

$21 = a \cdot 4^{2} + b (4) + c$

$41 = a \cdot 6^{2} + b (6) + c$

$c = 9 - 4 a - 2 b$

$21 = a \cdot 4^{2} + b (4) + c$

$41 = a \cdot 6^{2} + b (6) + c$

$c = 9 - 4 a - 2 b$

$21 = a \cdot 4^{2} + b (4) + c$

$41 = a \cdot 6^{2} + b (6) + c$

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

Replace all occurrences of $c$ in $21 = a \cdot 4^{2} + b (4) + c$ with $9 - 4 a - 2 b$ .

$21 = a \cdot 4^{2} + b (4) + 9 - 4 a - 2 b$

$c = 9 - 4 a - 2 b$

$41 = a \cdot 6^{2} + b (6) + c$

Simplify $21 = a \cdot 4^{2} + b (4) + 9 - 4 a - 2 b$ .

Simplify the left side.

Remove parentheses.

$21 = a \cdot 4^{2} + b (4) + 9 - 4 a - 2 b$

$c = 9 - 4 a - 2 b$

$41 = a \cdot 6^{2} + b (6) + c$

$21 = a \cdot 4^{2} + b (4) + 9 - 4 a - 2 b$

$c = 9 - 4 a - 2 b$

$41 = a \cdot 6^{2} + b (6) + c$

Simplify the right side.

Simplify $a \cdot 4^{2} + b (4) + 9 - 4 a - 2 b$ .

Simplify each term.

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

$21 = a \cdot 16 + b (4) + 9 - 4 a - 2 b$

$c = 9 - 4 a - 2 b$

$41 = a \cdot 6^{2} + b (6) + c$

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

$21 = 16 \cdot a + b (4) + 9 - 4 a - 2 b$

$c = 9 - 4 a - 2 b$

$41 = a \cdot 6^{2} + b (6) + c$

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

$21 = 16 a + 4 b + 9 - 4 a - 2 b$

$c = 9 - 4 a - 2 b$

$41 = a \cdot 6^{2} + b (6) + c$

$21 = 16 a + 4 b + 9 - 4 a - 2 b$

$c = 9 - 4 a - 2 b$

$41 = a \cdot 6^{2} + b (6) + c$

Simplify by adding terms.

Subtract $4 a$ from $16 a$ .

$21 = 12 a + 4 b + 9 - 2 b$

$c = 9 - 4 a - 2 b$

$41 = a \cdot 6^{2} + b (6) + c$

Subtract $2 b$ from $4 b$ .

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$41 = a \cdot 6^{2} + b (6) + c$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$41 = a \cdot 6^{2} + b (6) + c$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$41 = a \cdot 6^{2} + b (6) + c$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$41 = a \cdot 6^{2} + b (6) + c$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$41 = a \cdot 6^{2} + b (6) + c$

Replace all occurrences of $c$ in $41 = a \cdot 6^{2} + b (6) + c$ with $9 - 4 a - 2 b$ .

$41 = a \cdot 6^{2} + b (6) + 9 - 4 a - 2 b$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Simplify $41 = a \cdot 6^{2} + b (6) + 9 - 4 a - 2 b$ .

Simplify the left side.

Remove parentheses.

$41 = a \cdot 6^{2} + b (6) + 9 - 4 a - 2 b$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$41 = a \cdot 6^{2} + b (6) + 9 - 4 a - 2 b$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Simplify the right side.

Simplify $a \cdot 6^{2} + b (6) + 9 - 4 a - 2 b$ .

Simplify each term.

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

$41 = a \cdot 36 + b (6) + 9 - 4 a - 2 b$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

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

$41 = 36 \cdot a + b (6) + 9 - 4 a - 2 b$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

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

$41 = 36 a + 6 b + 9 - 4 a - 2 b$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$41 = 36 a + 6 b + 9 - 4 a - 2 b$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Simplify by adding terms.

Subtract $4 a$ from $36 a$ .

$41 = 32 a + 6 b + 9 - 2 b$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Subtract $2 b$ from $6 b$ .

$41 = 32 a + 4 b + 9$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$41 = 32 a + 4 b + 9$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$41 = 32 a + 4 b + 9$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$41 = 32 a + 4 b + 9$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$41 = 32 a + 4 b + 9$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$41 = 32 a + 4 b + 9$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Solve for $a$ in $41 = 32 a + 4 b + 9$ .

Rewrite the equation as $32 a + 4 b + 9 = 41$ .

$32 a + 4 b + 9 = 41$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

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

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

$32 a + 9 = 41 - 4 b$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Subtract $9$ from both sides of the equation.

$32 a = 41 - 4 b - 9$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Subtract $9$ from $41$ .

$32 a = - 4 b + 32$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$32 a = - 4 b + 32$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Divide each term in $32 a = - 4 b + 32$ by $32$ and simplify.

Divide each term in $32 a = - 4 b + 32$ by $32$ .

$\frac{32 a}{32} = \frac{- 4 b}{32} + \frac{32}{32}$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Simplify the left side.

Cancel the common factor of $32$ .

Cancel the common factor.

$\frac{32 a}{32} = \frac{- 4 b}{32} + \frac{32}{32}$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Divide $a$ by $1$ .

$a = \frac{- 4 b}{32} + \frac{32}{32}$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$a = \frac{- 4 b}{32} + \frac{32}{32}$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$a = \frac{- 4 b}{32} + \frac{32}{32}$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 4$ and $32$ .

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

$a = \frac{4 (- b)}{32} + \frac{32}{32}$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Cancel the common factors.

Factor $4$ out of $32$ .

$a = \frac{4 (- b)}{4 (8)} + \frac{32}{32}$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Cancel the common factor.

$a = \frac{4 (- b)}{4 \cdot 8} + \frac{32}{32}$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Rewrite the expression.

$a = \frac{- b}{8} + \frac{32}{32}$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$a = \frac{- b}{8} + \frac{32}{32}$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$a = \frac{- b}{8} + \frac{32}{32}$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Move the negative in front of the fraction.

$a = - \frac{b}{8} + \frac{32}{32}$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Divide $32$ by $32$ .

$a = - \frac{b}{8} + 1$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$a = - \frac{b}{8} + 1$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$a = - \frac{b}{8} + 1$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$a = - \frac{b}{8} + 1$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

$a = - \frac{b}{8} + 1$

$21 = 12 a + 2 b + 9$

$c = 9 - 4 a - 2 b$

Replace all occurrences of $a$ with $- \frac{b}{8} + 1$ in each equation.

Replace all occurrences of $a$ in $21 = 12 a + 2 b + 9$ with $- \frac{b}{8} + 1$ .

$21 = 12 (- \frac{b}{8} + 1) + 2 b + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Simplify the right side.

Simplify $12 (- \frac{b}{8} + 1) + 2 b + 9$ .

Simplify each term.

Apply the distributive property.

$21 = 12 (- \frac{b}{8}) + 12 \cdot 1 + 2 b + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{b}{8}$ into the numerator.

$21 = 12 (\frac{- b}{8}) + 12 \cdot 1 + 2 b + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Factor $4$ out of $12$ .

$21 = 4 (3) (\frac{- b}{8}) + 12 \cdot 1 + 2 b + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Factor $4$ out of $8$ .

$21 = 4 \cdot (3 (\frac{- b}{4 \cdot 2})) + 12 \cdot 1 + 2 b + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Cancel the common factor.

$21 = 4 \cdot (3 (\frac{- b}{4 \cdot 2})) + 12 \cdot 1 + 2 b + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Rewrite the expression.

$21 = 3 (\frac{- b}{2}) + 12 \cdot 1 + 2 b + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

$21 = 3 (\frac{- b}{2}) + 12 \cdot 1 + 2 b + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

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

$21 = \frac{3 (- b)}{2} + 12 \cdot 1 + 2 b + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Multiply $- 1$ by $3$ .

$21 = \frac{- 3 b}{2} + 12 \cdot 1 + 2 b + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Multiply $12$ by $1$ .

$21 = \frac{- 3 b}{2} + 12 + 2 b + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Move the negative in front of the fraction.

$21 = - \frac{3 b}{2} + 12 + 2 b + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

$21 = - \frac{3 b}{2} + 12 + 2 b + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

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

$21 = - \frac{3 b}{2} + 2 b \cdot \frac{2}{2} + 12 + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

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

$21 = - \frac{3 b}{2} + \frac{2 b \cdot 2}{2} + 12 + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Combine the numerators over the common denominator.

$21 = \frac{- 3 b + 2 b \cdot 2}{2} + 12 + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Find the common denominator.

Write $12$ as a fraction with denominator $1$ .

$21 = \frac{- 3 b + 2 b \cdot 2}{2} + \frac{12}{1} + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Multiply $\frac{12}{1}$ by $\frac{2}{2}$ .

$21 = \frac{- 3 b + 2 b \cdot 2}{2} + \frac{12}{1} \cdot \frac{2}{2} + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Multiply $\frac{12}{1}$ by $\frac{2}{2}$ .

$21 = \frac{- 3 b + 2 b \cdot 2}{2} + \frac{12 \cdot 2}{2} + 9$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

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

$21 = \frac{- 3 b + 2 b \cdot 2}{2} + \frac{12 \cdot 2}{2} + \frac{9}{1}$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Multiply $\frac{9}{1}$ by $\frac{2}{2}$ .

$21 = \frac{- 3 b + 2 b \cdot 2}{2} + \frac{12 \cdot 2}{2} + \frac{9}{1} \cdot \frac{2}{2}$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Multiply $\frac{9}{1}$ by $\frac{2}{2}$ .

$21 = \frac{- 3 b + 2 b \cdot 2}{2} + \frac{12 \cdot 2}{2} + \frac{9 \cdot 2}{2}$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

$21 = \frac{- 3 b + 2 b \cdot 2}{2} + \frac{12 \cdot 2}{2} + \frac{9 \cdot 2}{2}$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Combine the numerators over the common denominator.

$21 = \frac{- 3 b + 2 b \cdot 2 + 12 \cdot 2 + 9 \cdot 2}{2}$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Simplify each term.

Multiply $2$ by $2$ .

$21 = \frac{- 3 b + 4 b + 12 \cdot 2 + 9 \cdot 2}{2}$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Multiply $12$ by $2$ .

$21 = \frac{- 3 b + 4 b + 24 + 9 \cdot 2}{2}$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Multiply $9$ by $2$ .

$21 = \frac{- 3 b + 4 b + 24 + 18}{2}$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

$21 = \frac{- 3 b + 4 b + 24 + 18}{2}$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Simplify by adding terms.

Add $- 3 b$ and $4 b$ .

$21 = \frac{b + 24 + 18}{2}$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Add $24$ and $18$ .

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

$c = 9 - 4 a - 2 b$

Replace all occurrences of $a$ in $c = 9 - 4 a - 2 b$ with $- \frac{b}{8} + 1$ .

$c = 9 - 4 (- \frac{b}{8} + 1) - 2 b$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Simplify the right side.

Simplify $9 - 4 (- \frac{b}{8} + 1) - 2 b$ .

Simplify each term.

Apply the distributive property.

$c = 9 - 4 (- \frac{b}{8}) - 4 \cdot 1 - 2 b$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{b}{8}$ into the numerator.

$c = 9 - 4 \frac{- b}{8} - 4 \cdot 1 - 2 b$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Factor $4$ out of $- 4$ .

$c = 9 + 4 (- 1) (\frac{- b}{8}) - 4 \cdot 1 - 2 b$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Factor $4$ out of $8$ .

$c = 9 + 4 \cdot (- 1 \frac{- b}{4 \cdot 2}) - 4 \cdot 1 - 2 b$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Cancel the common factor.

$c = 9 + 4 \cdot (- 1 \frac{- b}{4 \cdot 2}) - 4 \cdot 1 - 2 b$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Rewrite the expression.

$c = 9 - 1 \frac{- b}{2} - 4 \cdot 1 - 2 b$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

$c = 9 - 1 \frac{- b}{2} - 4 \cdot 1 - 2 b$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Multiply $- 4$ by $1$ .

$c = 9 - 1 \frac{- b}{2} - 4 - 2 b$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Simplify each term.

Move the negative in front of the fraction.

$c = 9 - 1 (- \frac{b}{2}) - 4 - 2 b$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Multiply $- 1 (- \frac{b}{2})$ .

Multiply $- 1$ by $- 1$ .

$c = 9 + 1 (\frac{b}{2}) - 4 - 2 b$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Multiply $\frac{b}{2}$ by $1$ .

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

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

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

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

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

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

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

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Subtract $4$ from $9$ .

$c = \frac{b}{2} + 5 - 2 b$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

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

$c = \frac{b}{2} - 2 b \cdot \frac{2}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Simplify terms.

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

$c = \frac{b}{2} + \frac{- 2 b \cdot 2}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Combine the numerators over the common denominator.

$c = \frac{b - 2 b \cdot 2}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

$c = \frac{b - 2 b \cdot 2}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Simplify each term.

Simplify the numerator.

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

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

$c = \frac{b - 2 b \cdot 2}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Factor $b$ out of $b^{1}$ .

$c = \frac{b \cdot 1 - 2 b \cdot 2}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

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

$c = \frac{b \cdot 1 + b (- 2 \cdot 2)}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

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

$c = \frac{b (1 - 2 \cdot 2)}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

$c = \frac{b (1 - 2 \cdot 2)}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Multiply $- 2$ by $2$ .

$c = \frac{b (1 - 4)}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Subtract $4$ from $1$ .

$c = \frac{b \cdot - 3}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

$c = \frac{b \cdot - 3}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

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

$c = \frac{- 3 \cdot b}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Move the negative in front of the fraction.

$c = - \frac{3 b}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

$c = - \frac{3 b}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

$c = - \frac{3 b}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

$c = - \frac{3 b}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

$c = - \frac{3 b}{2} + 5$

$21 = \frac{b + 42}{2}$

$a = - \frac{b}{8} + 1$

Solve for $b$ in $21 = \frac{b + 42}{2}$ .

Rewrite the equation as $\frac{b + 42}{2} = 21$ .

$\frac{b + 42}{2} = 21$

$c = - \frac{3 b}{2} + 5$

$a = - \frac{b}{8} + 1$

Multiply both sides of the equation by $2$ .

$2 (\frac{b + 42}{2}) = 2 \cdot 21$

$c = - \frac{3 b}{2} + 5$

$a = - \frac{b}{8} + 1$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 (\frac{b + 42}{2}) = 2 \cdot 21$

$c = - \frac{3 b}{2} + 5$

$a = - \frac{b}{8} + 1$

Rewrite the expression.

$b + 42 = 2 \cdot 21$

$c = - \frac{3 b}{2} + 5$

$a = - \frac{b}{8} + 1$

$b + 42 = 2 \cdot 21$

$c = - \frac{3 b}{2} + 5$

$a = - \frac{b}{8} + 1$

$b + 42 = 2 \cdot 21$

$c = - \frac{3 b}{2} + 5$

$a = - \frac{b}{8} + 1$

Simplify the right side.

Multiply $2$ by $21$ .

$b + 42 = 42$

$c = - \frac{3 b}{2} + 5$

$a = - \frac{b}{8} + 1$

$b + 42 = 42$

$c = - \frac{3 b}{2} + 5$

$a = - \frac{b}{8} + 1$

$b + 42 = 42$

$c = - \frac{3 b}{2} + 5$

$a = - \frac{b}{8} + 1$

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

Subtract $42$ from both sides of the equation.

$b = 42 - 42$

$c = - \frac{3 b}{2} + 5$

$a = - \frac{b}{8} + 1$

Subtract $42$ from $42$ .

$b = 0$

$c = - \frac{3 b}{2} + 5$

$a = - \frac{b}{8} + 1$

$b = 0$

$c = - \frac{3 b}{2} + 5$

$a = - \frac{b}{8} + 1$

$b = 0$

$c = - \frac{3 b}{2} + 5$

$a = - \frac{b}{8} + 1$

Replace all occurrences of $b$ with $0$ in each equation.

Replace all occurrences of $b$ in $c = - \frac{3 b}{2} + 5$ with $0$ .

$c = - \frac{3 (0)}{2} + 5$

$b = 0$

$a = - \frac{b}{8} + 1$

Simplify the right side.

Simplify $- \frac{3 (0)}{2} + 5$ .

Simplify each term.

Cancel the common factor of $0$ and $2$ .

Factor $2$ out of $3 (0)$ .

$c = - \frac{2 (3 \cdot (0))}{2} + 5$

$b = 0$

$a = - \frac{b}{8} + 1$

Cancel the common factors.

Factor $2$ out of $2$ .

$c = - \frac{2 (3 \cdot (0))}{2 (1)} + 5$

$b = 0$

$a = - \frac{b}{8} + 1$

Cancel the common factor.

$c = - \frac{2 (3 \cdot (0))}{2 \cdot 1} + 5$

$b = 0$

$a = - \frac{b}{8} + 1$

Rewrite the expression.

$c = - \frac{3 \cdot (0)}{1} + 5$

$b = 0$

$a = - \frac{b}{8} + 1$

Divide $3 \cdot (0)$ by $1$ .

$c = - (3 \cdot (0)) + 5$

$b = 0$

$a = - \frac{b}{8} + 1$

$c = - (3 \cdot (0)) + 5$

$b = 0$

$a = - \frac{b}{8} + 1$

$c = - (3 \cdot (0)) + 5$

$b = 0$

$a = - \frac{b}{8} + 1$

Multiply $- (3 \cdot (0))$ .

Multiply $3$ by $0$ .

$c = - 0 + 5$

$b = 0$

$a = - \frac{b}{8} + 1$

Multiply $- 1$ by $0$ .

$c = 0 + 5$

$b = 0$

$a = - \frac{b}{8} + 1$

$c = 0 + 5$

$b = 0$

$a = - \frac{b}{8} + 1$

$c = 0 + 5$

$b = 0$

$a = - \frac{b}{8} + 1$

Add $0$ and $5$ .

$c = 5$

$b = 0$

$a = - \frac{b}{8} + 1$

$c = 5$

$b = 0$

$a = - \frac{b}{8} + 1$

$c = 5$

$b = 0$

$a = - \frac{b}{8} + 1$

Replace all occurrences of $b$ in $a = - \frac{b}{8} + 1$ with $0$ .

$a = - \frac{0}{8} + 1$

$c = 5$

$b = 0$

Simplify the right side.

Simplify $- \frac{0}{8} + 1$ .

Simplify each term.

Divide $0$ by $8$ .

$a = - 0 + 1$

$c = 5$

$b = 0$

Multiply $- 1$ by $0$ .

$a = 0 + 1$

$c = 5$

$b = 0$

$a = 0 + 1$

$c = 5$

$b = 0$

Add $0$ and $1$ .

$a = 1$

$c = 5$

$b = 0$

$a = 1$

$c = 5$

$b = 0$

$a = 1$

$c = 5$

$b = 0$

$a = 1$

$c = 5$

$b = 0$

List all of the solutions.

$a = 1, c = 5, b = 0$

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 $a = 1$ , $b = 0$ , $c = 5$ , and $x = 2$ .

Simplify each term.

Multiply ${(2)}^{2}$ by $1$ .

$y = {(2)}^{2} + (0) \cdot (2) + 5$

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

$y = 4 + (0) \cdot (2) + 5$

Multiply $0$ by $2$ .

$y = 4 + 0 + 5$

Simplify by adding numbers.

Add $4$ and $0$ .

$y = 4 + 5$

Add $4$ and $5$ .

$y = 9$

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 $a = 1$ , $b = 0$ , $c = 5$ , and $x = 4$ .

Simplify each term.

Multiply ${(4)}^{2}$ by $1$ .

$y = {(4)}^{2} + (0) \cdot (4) + 5$

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

$y = 16 + (0) \cdot (4) + 5$

Multiply $0$ by $4$ .

$y = 16 + 0 + 5$

Simplify by adding numbers.

Add $16$ and $0$ .

$y = 16 + 5$

Add $16$ and $5$ .

$y = 21$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 4$ . This check passes since $y = 21$ and $y = 21$ .

$21 = 21$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 1$ , $b = 0$ , $c = 5$ , and $x = 6$ .

Simplify each term.

Multiply ${(6)}^{2}$ by $1$ .

$y = {(6)}^{2} + (0) \cdot (6) + 5$

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

$y = 36 + (0) \cdot (6) + 5$

Multiply $0$ by $6$ .

$y = 36 + 0 + 5$

Simplify by adding numbers.

Add $36$ and $0$ .

$y = 36 + 5$

Add $36$ and $5$ .

$y = 41$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 6$ . This check passes since $y = 41$ and $y = 41$ .

$41 = 41$

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 $y = x^{2} + 5$ .

$y = x^{2} + 5$

Find the Function Rule table[[x,y],[2,9],[4,21],[6,41]]

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