Find the Function Rule table[[x,y],[-2,16],[1,4],[2,0]]

$\begin{array}{c} x & y \\ - 2 & 16 \\ 1 & 4 \\ 2 & 0 \end{array}$

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 $y = a x + b$ .

$16 = a (- 2) + b 4 = a (1) + b 0 = a (2) + b$

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

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

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

$a + b = 4$

$16 = a (- 2) + b$

$0 = a (2) + b$

Subtract $b$ from both sides of the equation.

$a = 4 - b$

$16 = a (- 2) + b$

$0 = a (2) + b$

$a = 4 - b$

$16 = a (- 2) + b$

$0 = a (2) + b$

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

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

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

$a = 4 - b$

$0 = a (2) + b$

Simplify the right side.

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

Simplify each term.

Apply the distributive property.

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

$a = 4 - b$

$0 = a (2) + b$

Multiply $4$ by $- 2$ .

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

$a = 4 - b$

$0 = a (2) + b$

Multiply $- 2$ by $- 1$ .

$16 = - 8 + 2 b + b$

$a = 4 - b$

$0 = a (2) + b$

$16 = - 8 + 2 b + b$

$a = 4 - b$

$0 = a (2) + b$

Add $2 b$ and $b$ .

$16 = - 8 + 3 b$

$a = 4 - b$

$0 = a (2) + b$

$16 = - 8 + 3 b$

$a = 4 - b$

$0 = a (2) + b$

$16 = - 8 + 3 b$

$a = 4 - b$

$0 = a (2) + b$

Replace all occurrences of $a$ in $0 = a (2) + b$ with $4 - b$ .

$0 = (4 - b) (2) + b$

$16 = - 8 + 3 b$

$a = 4 - b$

Simplify the right side.

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

Simplify each term.

Apply the distributive property.

$0 = 4 \cdot 2 - b \cdot 2 + b$

$16 = - 8 + 3 b$

$a = 4 - b$

Multiply $4$ by $2$ .

$0 = 8 - b \cdot 2 + b$

$16 = - 8 + 3 b$

$a = 4 - b$

Multiply $2$ by $- 1$ .

$0 = 8 - 2 b + b$

$16 = - 8 + 3 b$

$a = 4 - b$

$0 = 8 - 2 b + b$

$16 = - 8 + 3 b$

$a = 4 - b$

Add $- 2 b$ and $b$ .

$0 = 8 - b$

$16 = - 8 + 3 b$

$a = 4 - b$

$0 = 8 - b$

$16 = - 8 + 3 b$

$a = 4 - b$

$0 = 8 - b$

$16 = - 8 + 3 b$

$a = 4 - b$

$0 = 8 - b$

$16 = - 8 + 3 b$

$a = 4 - b$

Solve for $b$ in $0 = 8 - b$ .

Rewrite the equation as $8 - b = 0$ .

$8 - b = 0$

$16 = - 8 + 3 b$

$a = 4 - b$

Subtract $8$ from both sides of the equation.

$- b = - 8$

$16 = - 8 + 3 b$

$a = 4 - b$

Divide each term in $- b = - 8$ by $- 1$ and simplify.

Divide each term in $- b = - 8$ by $- 1$ .

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

$16 = - 8 + 3 b$

$a = 4 - b$

Simplify the left side.

Dividing two negative values results in a positive value.

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

$16 = - 8 + 3 b$

$a = 4 - b$

Divide $b$ by $1$ .

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

$16 = - 8 + 3 b$

$a = 4 - b$

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

$16 = - 8 + 3 b$

$a = 4 - b$

Simplify the right side.

Divide $- 8$ by $- 1$ .

$b = 8$

$16 = - 8 + 3 b$

$a = 4 - b$

$b = 8$

$16 = - 8 + 3 b$

$a = 4 - b$

$b = 8$

$16 = - 8 + 3 b$

$a = 4 - b$

$b = 8$

$16 = - 8 + 3 b$

$a = 4 - b$

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

Replace all occurrences of $b$ in $16 = - 8 + 3 b$ with $8$ .

$16 = - 8 + 3 (8)$

$b = 8$

$a = 4 - b$

Simplify the right side.

Simplify $- 8 + 3 (8)$ .

Multiply $3$ by $8$ .

$16 = - 8 + 24$

$b = 8$

$a = 4 - b$

Add $- 8$ and $24$ .

$16 = 16$

$b = 8$

$a = 4 - b$

$16 = 16$

$b = 8$

$a = 4 - b$

$16 = 16$

$b = 8$

$a = 4 - b$

Replace all occurrences of $b$ in $a = 4 - b$ with $8$ .

$a = 4 - (8)$

$16 = 16$

$b = 8$

Simplify the right side.

Simplify $4 - (8)$ .

Multiply $- 1$ by $8$ .

$a = 4 - 8$

$16 = 16$

$b = 8$

Subtract $8$ from $4$ .

$a = - 4$

$16 = 16$

$b = 8$

$a = - 4$

$16 = 16$

$b = 8$

$a = - 4$

$16 = 16$

$b = 8$

$a = - 4$

$16 = 16$

$b = 8$

Remove any equations from the system that are always true.

$a = - 4$

$b = 8$

List all of the solutions.

$a = - 4, b = 8$

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

Calculate the value of $y$ when $a = - 4$ , $b = 8$ , and $x = - 2$ .

Multiply $- 4$ by $- 2$ .

$y = 8 + 8$

Add $8$ and $8$ .

$y = 16$

If the table has a linear 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 $y$ when $a = - 4$ , $b = 8$ , and $x = 1$ .

Multiply $- 4$ by $1$ .

$y = - 4 + 8$

Add $- 4$ and $8$ .

$y = 4$

If the table has a linear 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 $y$ when $a = - 4$ , $b = 8$ , and $x = 2$ .

Multiply $- 4$ by $2$ .

$y = - 8 + 8$

Add $- 8$ and $8$ .

$y = 0$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 0$ and $y = 0$ .

$0 = 0$

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

The function is linear

Step 2

Since all $y = y$ , the function is linear and follows the form $y = - 4 x + 8$ .

$y = - 4 x + 8$

Find the Function Rule table[[x,y],[-2,16],[1,4],[2,0]]

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