Find the Function Rule table[[x,y],[1,4],[2,16],[3,64]]

$\begin{array}{c} x & y \\ 1 & 4 \\ 2 & 16 \\ 3 & 64 \end{array}$

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 $y = a x + b$ .

$4 = a (1) + b 16 = a (2) + b 64 = a (3) + b$

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

Solve for $a$ in $4 = a + b$ .

Rewrite the equation as $a + b = 4$ .

$a + b = 4$

$16 = a (2) + b$

$64 = a (3) + b$

Subtract $b$ from both sides of the equation.

$a = 4 - b$

$16 = a (2) + b$

$64 = a (3) + b$

$a = 4 - b$

$16 = a (2) + b$

$64 = a (3) + b$

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

Replace all occurrences of $a$ in $16 = a (2) + b$ with $4 - b$ .

$16 = (4 - b) (2) + b$

$a = 4 - b$

$64 = a (3) + b$

Simplify the right side.

Simplify $(4 - b) (2) + b$ .

Simplify each term.

Apply the distributive property.

$16 = 4 \cdot 2 - b \cdot 2 + b$

$a = 4 - b$

$64 = a (3) + b$

Multiply $4$ by $2$ .

$16 = 8 - b \cdot 2 + b$

$a = 4 - b$

$64 = a (3) + b$

Multiply $2$ by $- 1$ .

$16 = 8 - 2 b + b$

$a = 4 - b$

$64 = a (3) + b$

$16 = 8 - 2 b + b$

$a = 4 - b$

$64 = a (3) + b$

Add $- 2 b$ and $b$ .

$16 = 8 - b$

$a = 4 - b$

$64 = a (3) + b$

$16 = 8 - b$

$a = 4 - b$

$64 = a (3) + b$

$16 = 8 - b$

$a = 4 - b$

$64 = a (3) + b$

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

$64 = (4 - b) (3) + b$

$16 = 8 - b$

$a = 4 - b$

Simplify the right side.

Simplify $(4 - b) (3) + b$ .

Simplify each term.

Apply the distributive property.

$64 = 4 \cdot 3 - b \cdot 3 + b$

$16 = 8 - b$

$a = 4 - b$

Multiply $4$ by $3$ .

$64 = 12 - b \cdot 3 + b$

$16 = 8 - b$

$a = 4 - b$

Multiply $3$ by $- 1$ .

$64 = 12 - 3 b + b$

$16 = 8 - b$

$a = 4 - b$

$64 = 12 - 3 b + b$

$16 = 8 - b$

$a = 4 - b$

Add $- 3 b$ and $b$ .

$64 = 12 - 2 b$

$16 = 8 - b$

$a = 4 - b$

$64 = 12 - 2 b$

$16 = 8 - b$

$a = 4 - b$

$64 = 12 - 2 b$

$16 = 8 - b$

$a = 4 - b$

$64 = 12 - 2 b$

$16 = 8 - b$

$a = 4 - b$

Solve for $b$ in $64 = 12 - 2 b$ .

Rewrite the equation as $12 - 2 b = 64$ .

$12 - 2 b = 64$

$16 = 8 - b$

$a = 4 - b$

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

Subtract $12$ from both sides of the equation.

$- 2 b = 64 - 12$

$16 = 8 - b$

$a = 4 - b$

Subtract $12$ from $64$ .

$- 2 b = 52$

$16 = 8 - b$

$a = 4 - b$

$- 2 b = 52$

$16 = 8 - b$

$a = 4 - b$

Divide each term in $- 2 b = 52$ by $- 2$ and simplify.

Divide each term in $- 2 b = 52$ by $- 2$ .

$\frac{- 2 b}{- 2} = \frac{52}{- 2}$

$16 = 8 - b$

$a = 4 - b$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 b}{- 2} = \frac{52}{- 2}$

$16 = 8 - b$

$a = 4 - b$

Divide $b$ by $1$ .

$b = \frac{52}{- 2}$

$16 = 8 - b$

$a = 4 - b$

$b = \frac{52}{- 2}$

$16 = 8 - b$

$a = 4 - b$

$b = \frac{52}{- 2}$

$16 = 8 - b$

$a = 4 - b$

Simplify the right side.

Divide $52$ by $- 2$ .

$b = - 26$

$16 = 8 - b$

$a = 4 - b$

$b = - 26$

$16 = 8 - b$

$a = 4 - b$

$b = - 26$

$16 = 8 - b$

$a = 4 - b$

$b = - 26$

$16 = 8 - b$

$a = 4 - b$

Replace all occurrences of $b$ with $- 26$ in each equation.

Replace all occurrences of $b$ in $16 = 8 - b$ with $- 26$ .

$16 = 8 - (- 26)$

$b = - 26$

$a = 4 - b$

Simplify the right side.

Simplify $8 - (- 26)$ .

Multiply $- 1$ by $- 26$ .

$16 = 8 + 26$

$b = - 26$

$a = 4 - b$

Add $8$ and $26$ .

$16 = 34$

$b = - 26$

$a = 4 - b$

$16 = 34$

$b = - 26$

$a = 4 - b$

$16 = 34$

$b = - 26$

$a = 4 - b$

Replace all occurrences of $b$ in $a = 4 - b$ with $- 26$ .

$a = 4 - (- 26)$

$16 = 34$

$b = - 26$

Simplify the right side.

Simplify $4 - (- 26)$ .

Multiply $- 1$ by $- 26$ .

$a = 4 + 26$

$16 = 34$

$b = - 26$

Add $4$ and $26$ .

$a = 30$

$16 = 34$

$b = - 26$

$a = 30$

$16 = 34$

$b = - 26$

$a = 30$

$16 = 34$

$b = - 26$

$a = 30$

$16 = 34$

$b = - 26$

Since $16 = 34$ 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 $a$ in $4 = a + b + c$ .

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

$a + b + c = 4$

$16 = a \cdot 2^{2} + b (2) + c$

$64 = a \cdot 3^{2} + b (3) + c$

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

Subtract $b$ from both sides of the equation.

$a + c = 4 - b$

$16 = a \cdot 2^{2} + b (2) + c$

$64 = a \cdot 3^{2} + b (3) + c$

Subtract $c$ from both sides of the equation.

$a = 4 - b - c$

$16 = a \cdot 2^{2} + b (2) + c$

$64 = a \cdot 3^{2} + b (3) + c$

$a = 4 - b - c$

$16 = a \cdot 2^{2} + b (2) + c$

$64 = a \cdot 3^{2} + b (3) + c$

$a = 4 - b - c$

$16 = a \cdot 2^{2} + b (2) + c$

$64 = a \cdot 3^{2} + b (3) + c$

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

Replace all occurrences of $a$ in $16 = a \cdot 2^{2} + b (2) + c$ with $4 - b - c$ .

$16 = (4 - b - c) \cdot 2^{2} + b (2) + c$

$a = 4 - b - c$

$64 = a \cdot 3^{2} + b (3) + c$

Simplify the right side.

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

Simplify each term.

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

$16 = (4 - b - c) \cdot 4 + b (2) + c$

$a = 4 - b - c$

$64 = a \cdot 3^{2} + b (3) + c$

Apply the distributive property.

$16 = 4 \cdot 4 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = 4 - b - c$

$64 = a \cdot 3^{2} + b (3) + c$

Simplify.

Multiply $4$ by $4$ .

$16 = 16 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = 4 - b - c$

$64 = a \cdot 3^{2} + b (3) + c$

Multiply $4$ by $- 1$ .

$16 = 16 - 4 b - c \cdot 4 + b (2) + c$

$a = 4 - b - c$

$64 = a \cdot 3^{2} + b (3) + c$

Multiply $4$ by $- 1$ .

$16 = 16 - 4 b - 4 c + b (2) + c$

$a = 4 - b - c$

$64 = a \cdot 3^{2} + b (3) + c$

$16 = 16 - 4 b - 4 c + b (2) + c$

$a = 4 - b - c$

$64 = a \cdot 3^{2} + b (3) + c$

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

$16 = 16 - 4 b - 4 c + 2 b + c$

$a = 4 - b - c$

$64 = a \cdot 3^{2} + b (3) + c$

$16 = 16 - 4 b - 4 c + 2 b + c$

$a = 4 - b - c$

$64 = a \cdot 3^{2} + b (3) + c$

Simplify by adding terms.

Add $- 4 b$ and $2 b$ .

$16 = 16 - 2 b - 4 c + c$

$a = 4 - b - c$

$64 = a \cdot 3^{2} + b (3) + c$

Add $- 4 c$ and $c$ .

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$64 = a \cdot 3^{2} + b (3) + c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$64 = a \cdot 3^{2} + b (3) + c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$64 = a \cdot 3^{2} + b (3) + c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$64 = a \cdot 3^{2} + b (3) + c$

Replace all occurrences of $a$ in $64 = a \cdot 3^{2} + b (3) + c$ with $4 - b - c$ .

$64 = (4 - b - c) \cdot 3^{2} + b (3) + c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Simplify the right side.

Simplify $(4 - b - c) \cdot 3^{2} + b (3) + c$ .

Simplify each term.

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

$64 = (4 - b - c) \cdot 9 + b (3) + c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Apply the distributive property.

$64 = 4 \cdot 9 - b \cdot 9 - c \cdot 9 + b (3) + c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Simplify.

Multiply $4$ by $9$ .

$64 = 36 - b \cdot 9 - c \cdot 9 + b (3) + c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Multiply $9$ by $- 1$ .

$64 = 36 - 9 b - c \cdot 9 + b (3) + c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Multiply $9$ by $- 1$ .

$64 = 36 - 9 b - 9 c + b (3) + c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$64 = 36 - 9 b - 9 c + b (3) + c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

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

$64 = 36 - 9 b - 9 c + 3 b + c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$64 = 36 - 9 b - 9 c + 3 b + c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Simplify by adding terms.

Add $- 9 b$ and $3 b$ .

$64 = 36 - 6 b - 9 c + c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Add $- 9 c$ and $c$ .

$64 = 36 - 6 b - 8 c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$64 = 36 - 6 b - 8 c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$64 = 36 - 6 b - 8 c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$64 = 36 - 6 b - 8 c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$64 = 36 - 6 b - 8 c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Solve for $b$ in $64 = 36 - 6 b - 8 c$ .

Rewrite the equation as $36 - 6 b - 8 c = 64$ .

$36 - 6 b - 8 c = 64$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

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

Subtract $36$ from both sides of the equation.

$- 6 b - 8 c = 64 - 36$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Add $8 c$ to both sides of the equation.

$- 6 b = 64 - 36 + 8 c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Subtract $36$ from $64$ .

$- 6 b = 28 + 8 c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$- 6 b = 28 + 8 c$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Divide each term in $- 6 b = 28 + 8 c$ by $- 6$ and simplify.

Divide each term in $- 6 b = 28 + 8 c$ by $- 6$ .

$\frac{- 6 b}{- 6} = \frac{28}{- 6} + \frac{8 c}{- 6}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 b}{- 6} = \frac{28}{- 6} + \frac{8 c}{- 6}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Divide $b$ by $1$ .

$b = \frac{28}{- 6} + \frac{8 c}{- 6}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$b = \frac{28}{- 6} + \frac{8 c}{- 6}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$b = \frac{28}{- 6} + \frac{8 c}{- 6}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Simplify the right side.

Simplify each term.

Cancel the common factor of $28$ and $- 6$ .

Factor $2$ out of $28$ .

$b = \frac{2 (14)}{- 6} + \frac{8 c}{- 6}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Cancel the common factors.

Factor $2$ out of $- 6$ .

$b = \frac{2 \cdot 14}{2 \cdot - 3} + \frac{8 c}{- 6}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Cancel the common factor.

$b = \frac{2 \cdot 14}{2 \cdot - 3} + \frac{8 c}{- 6}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Rewrite the expression.

$b = \frac{14}{- 3} + \frac{8 c}{- 6}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$b = \frac{14}{- 3} + \frac{8 c}{- 6}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$b = \frac{14}{- 3} + \frac{8 c}{- 6}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Move the negative in front of the fraction.

$b = - \frac{14}{3} + \frac{8 c}{- 6}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Cancel the common factor of $8$ and $- 6$ .

Factor $2$ out of $8 c$ .

$b = - \frac{14}{3} + \frac{2 (4 c)}{- 6}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Cancel the common factors.

Factor $2$ out of $- 6$ .

$b = - \frac{14}{3} + \frac{2 (4 c)}{2 (- 3)}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Cancel the common factor.

$b = - \frac{14}{3} + \frac{2 (4 c)}{2 \cdot - 3}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Rewrite the expression.

$b = - \frac{14}{3} + \frac{4 c}{- 3}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$b = - \frac{14}{3} + \frac{4 c}{- 3}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$b = - \frac{14}{3} + \frac{4 c}{- 3}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Move the negative in front of the fraction.

$b = - \frac{14}{3} - \frac{4 c}{3}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$16 = 16 - 2 b - 3 c$

$a = 4 - b - c$

Replace all occurrences of $b$ with $- \frac{14}{3} - \frac{4 c}{3}$ in each equation.

Replace all occurrences of $b$ in $16 = 16 - 2 b - 3 c$ with $- \frac{14}{3} - \frac{4 c}{3}$ .

$16 = 16 - 2 (- \frac{14}{3} - \frac{4 c}{3}) - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Simplify the right side.

Simplify $16 - 2 (- \frac{14}{3} - \frac{4 c}{3}) - 3 c$ .

Simplify each term.

Apply the distributive property.

$16 = 16 - 2 (- \frac{14}{3}) - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Multiply $- 2 (- \frac{14}{3})$ .

Multiply $- 1$ by $- 2$ .

$16 = 16 + 2 (\frac{14}{3}) - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Combine $2$ and $\frac{14}{3}$ .

$16 = 16 + \frac{2 \cdot 14}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Multiply $2$ by $14$ .

$16 = 16 + \frac{28}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

$16 = 16 + \frac{28}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Multiply $- 2 (- \frac{4 c}{3})$ .

Multiply $- 1$ by $- 2$ .

$16 = 16 + \frac{28}{3} + 2 (\frac{4 c}{3}) - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Combine $2$ and $\frac{4 c}{3}$ .

$16 = 16 + \frac{28}{3} + \frac{2 (4 c)}{3} - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Multiply $4$ by $2$ .

$16 = 16 + \frac{28}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

$16 = 16 + \frac{28}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

$16 = 16 + \frac{28}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

To write $16$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$16 = 16 \cdot \frac{3}{3} + \frac{28}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

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

$16 = \frac{16 \cdot 3}{3} + \frac{28}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Combine the numerators over the common denominator.

$16 = \frac{16 \cdot 3 + 28}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Simplify the numerator.

Multiply $16$ by $3$ .

$16 = \frac{48 + 28}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Add $48$ and $28$ .

$16 = \frac{76}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

$16 = \frac{76}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

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

$16 = \frac{76}{3} + \frac{8 c}{3} - 3 c \cdot \frac{3}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Combine $- 3 c$ and $\frac{3}{3}$ .

$16 = \frac{76}{3} + \frac{8 c}{3} + \frac{- 3 c \cdot 3}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Combine the numerators over the common denominator.

$16 = \frac{76}{3} + \frac{8 c - 3 c \cdot 3}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Combine the numerators over the common denominator.

$16 = \frac{76 + 8 c - 3 c \cdot 3}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Multiply $3$ by $- 3$ .

$16 = \frac{76 + 8 c - 9 c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Subtract $9 c$ from $8 c$ .

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 - b - c$

Replace all occurrences of $b$ in $a = 4 - b - c$ with $- \frac{14}{3} - \frac{4 c}{3}$ .

$a = 4 - (- \frac{14}{3} - \frac{4 c}{3}) - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Simplify the right side.

Simplify $4 - (- \frac{14}{3} - \frac{4 c}{3}) - c$ .

Simplify each term.

Apply the distributive property.

$a = 4 + \frac{14}{3} + \frac{4 c}{3} - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Multiply $- - \frac{14}{3}$ .

Multiply $- 1$ by $- 1$ .

$a = 4 + 1 (\frac{14}{3}) + \frac{4 c}{3} - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Multiply $\frac{14}{3}$ by $1$ .

$a = 4 + \frac{14}{3} + \frac{4 c}{3} - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 + \frac{14}{3} + \frac{4 c}{3} - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Multiply $- - \frac{4 c}{3}$ .

Multiply $- 1$ by $- 1$ .

$a = 4 + \frac{14}{3} + 1 (\frac{4 c}{3}) - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Multiply $\frac{4 c}{3}$ by $1$ .

$a = 4 + \frac{14}{3} + \frac{4 c}{3} - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 + \frac{14}{3} + \frac{4 c}{3} - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 4 + \frac{14}{3} + \frac{4 c}{3} - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

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

$a = 4 \cdot \frac{3}{3} + \frac{14}{3} + \frac{4 c}{3} - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

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

$a = \frac{4 \cdot 3}{3} + \frac{14}{3} + \frac{4 c}{3} - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Combine the numerators over the common denominator.

$a = \frac{4 \cdot 3 + 14}{3} + \frac{4 c}{3} - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Simplify the numerator.

Multiply $4$ by $3$ .

$a = \frac{12 + 14}{3} + \frac{4 c}{3} - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Add $12$ and $14$ .

$a = \frac{26}{3} + \frac{4 c}{3} - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = \frac{26}{3} + \frac{4 c}{3} - c$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

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

$a = \frac{26}{3} + \frac{4 c}{3} - c \cdot \frac{3}{3}$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Combine $- c$ and $\frac{3}{3}$ .

$a = \frac{26}{3} + \frac{4 c}{3} + \frac{- c \cdot 3}{3}$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Combine the numerators over the common denominator.

$a = \frac{26}{3} + \frac{4 c - c \cdot 3}{3}$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Combine the numerators over the common denominator.

$a = \frac{26 + 4 c - c \cdot 3}{3}$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Multiply $3$ by $- 1$ .

$a = \frac{26 + 4 c - 3 c}{3}$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Subtract $3 c$ from $4 c$ .

$a = \frac{26 + c}{3}$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = \frac{26 + c}{3}$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = \frac{26 + c}{3}$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = \frac{26 + c}{3}$

$16 = \frac{76 - c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Solve for $c$ in $16 = \frac{76 - c}{3}$ .

Rewrite the equation as $\frac{76 - c}{3} = 16$ .

$\frac{76 - c}{3} = 16$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Multiply both sides by $3$ .

$\frac{76 - c}{3} \cdot 3 = 16 \cdot 3$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Simplify.

Simplify the left side.

Simplify $\frac{76 - c}{3} \cdot 3$ .

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{76 - c}{3} \cdot 3 = 16 \cdot 3$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Rewrite the expression.

$76 - c = 16 \cdot 3$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$76 - c = 16 \cdot 3$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Reorder $76$ and $- c$ .

$- c + 76 = 16 \cdot 3$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$- c + 76 = 16 \cdot 3$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$- c + 76 = 16 \cdot 3$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Simplify the right side.

Multiply $16$ by $3$ .

$- c + 76 = 48$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$- c + 76 = 48$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$- c + 76 = 48$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Solve for $c$ .

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

Subtract $76$ from both sides of the equation.

$- c = 48 - 76$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Subtract $76$ from $48$ .

$- c = - 28$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$- c = - 28$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Divide each term in $- c = - 28$ by $- 1$ and simplify.

Divide each term in $- c = - 28$ by $- 1$ .

$\frac{- c}{- 1} = \frac{- 28}{- 1}$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{c}{1} = \frac{- 28}{- 1}$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Divide $c$ by $1$ .

$c = \frac{- 28}{- 1}$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$c = \frac{- 28}{- 1}$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Simplify the right side.

Divide $- 28$ by $- 1$ .

$c = 28$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$c = 28$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$c = 28$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$c = 28$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$c = 28$

$a = \frac{26 + c}{3}$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Replace all occurrences of $c$ with $28$ in each equation.

Replace all occurrences of $c$ in $a = \frac{26 + c}{3}$ with $28$ .

$a = \frac{26 + 28}{3}$

$c = 28$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Simplify $a = \frac{26 + 28}{3}$ .

Simplify the left side.

Remove parentheses.

$a = \frac{26 + 28}{3}$

$c = 28$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = \frac{26 + 28}{3}$

$c = 28$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Simplify the right side.

Simplify $\frac{26 + 28}{3}$ .

Add $26$ and $28$ .

$a = \frac{54}{3}$

$c = 28$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Divide $54$ by $3$ .

$a = 18$

$c = 28$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 18$

$c = 28$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 18$

$c = 28$

$b = - \frac{14}{3} - \frac{4 c}{3}$

$a = 18$

$c = 28$

$b = - \frac{14}{3} - \frac{4 c}{3}$

Replace all occurrences of $c$ in $b = - \frac{14}{3} - \frac{4 c}{3}$ with $28$ .

$b = - \frac{14}{3} - \frac{4 (28)}{3}$

$a = 18$

$c = 28$

Simplify the right side.

Simplify $- \frac{14}{3} - \frac{4 (28)}{3}$ .

Combine the numerators over the common denominator.

$b = \frac{- 14 - 4 \cdot 28}{3}$

$a = 18$

$c = 28$

Simplify the expression.

Multiply $- 4$ by $28$ .

$b = \frac{- 14 - 112}{3}$

$a = 18$

$c = 28$

Subtract $112$ from $- 14$ .

$b = \frac{- 126}{3}$

$a = 18$

$c = 28$

Divide $- 126$ by $3$ .

$b = - 42$

$a = 18$

$c = 28$

$b = - 42$

$a = 18$

$c = 28$

$b = - 42$

$a = 18$

$c = 28$

$b = - 42$

$a = 18$

$c = 28$

$b = - 42$

$a = 18$

$c = 28$

List all of the solutions.

$b = - 42, a = 18, c = 28$

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 $a = 18$ , $b = - 42$ , $c = 28$ , and $x = 1$ .

Simplify each term.

One to any power is one.

$y = 18 \cdot 1 + (- 42) \cdot (1) + 28$

Multiply $18$ by $1$ .

$y = 18 + (- 42) \cdot (1) + 28$

Multiply $- 42$ by $1$ .

$y = 18 - 42 + 28$

Simplify by adding and subtracting.

Subtract $42$ from $18$ .

$y = - 24 + 28$

Add $- 24$ and $28$ .

$y = 4$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = 4$ and $y = 4$ .

$4 = 4$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 18$ , $b = - 42$ , $c = 28$ , and $x = 2$ .

Simplify each term.

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

$y = 18 \cdot 4 + (- 42) \cdot (2) + 28$

Multiply $18$ by $4$ .

$y = 72 + (- 42) \cdot (2) + 28$

Multiply $- 42$ by $2$ .

$y = 72 - 84 + 28$

Simplify by adding and subtracting.

Subtract $84$ from $72$ .

$y = - 12 + 28$

Add $- 12$ and $28$ .

$y = 16$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 16$ and $y = 16$ .

$16 = 16$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 18$ , $b = - 42$ , $c = 28$ , and $x = 3$ .

Simplify each term.

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

$y = 18 \cdot 9 + (- 42) \cdot (3) + 28$

Multiply $18$ by $9$ .

$y = 162 + (- 42) \cdot (3) + 28$

Multiply $- 42$ by $3$ .

$y = 162 - 126 + 28$

Simplify by adding and subtracting.

Subtract $126$ from $162$ .

$y = 36 + 28$

Add $36$ and $28$ .

$y = 64$

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

$64 = 64$

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 $y = 18 x^{2} - 42 x + 28$ .

$y = 18 x^{2} - 42 x + 28$

Find the Function Rule table[[x,y],[1,4],[2,16],[3,64]]

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