Find the Function Rule table[[x,y],[0,14],[2,6],[4,-2]]

$\begin{array}{c} x & y \\ 0 & 14 \\ 2 & 6 \\ 4 & - 2 \end{array}$

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 $y = a x + b$ .

$14 = a (0) + b 6 = a (2) + b - 2 = a (4) + b$

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

Rewrite the equation as $b = 14$ .

$b = 14$

$6 = a (2) + b$

$- 2 = a (4) + b$

Replace all occurrences of $b$ with $14$ in each equation.

Replace all occurrences of $b$ in $6 = a (2) + b$ with $14$ .

$6 = a (2) + 14$

$b = 14$

$- 2 = a (4) + b$

Simplify $6 = a (2) + 14$ .

Simplify the left side.

Remove parentheses.

$6 = a (2) + 14$

$b = 14$

$- 2 = a (4) + b$

$6 = a (2) + 14$

$b = 14$

$- 2 = a (4) + b$

Simplify the right side.

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

$6 = 2 a + 14$

$b = 14$

$- 2 = a (4) + b$

$6 = 2 a + 14$

$b = 14$

$- 2 = a (4) + b$

$6 = 2 a + 14$

$b = 14$

$- 2 = a (4) + b$

Replace all occurrences of $b$ in $- 2 = a (4) + b$ with $14$ .

$- 2 = a (4) + 14$

$6 = 2 a + 14$

$b = 14$

Simplify $- 2 = a (4) + 14$ .

Simplify the left side.

Remove parentheses.

$- 2 = a (4) + 14$

$6 = 2 a + 14$

$b = 14$

$- 2 = a (4) + 14$

$6 = 2 a + 14$

$b = 14$

Simplify the right side.

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

$- 2 = 4 a + 14$

$6 = 2 a + 14$

$b = 14$

$- 2 = 4 a + 14$

$6 = 2 a + 14$

$b = 14$

$- 2 = 4 a + 14$

$6 = 2 a + 14$

$b = 14$

$- 2 = 4 a + 14$

$6 = 2 a + 14$

$b = 14$

Solve for $a$ in $- 2 = 4 a + 14$ .

Rewrite the equation as $4 a + 14 = - 2$ .

$4 a + 14 = - 2$

$6 = 2 a + 14$

$b = 14$

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

Subtract $14$ from both sides of the equation.

$4 a = - 2 - 14$

$6 = 2 a + 14$

$b = 14$

Subtract $14$ from $- 2$ .

$4 a = - 16$

$6 = 2 a + 14$

$b = 14$

$4 a = - 16$

$6 = 2 a + 14$

$b = 14$

Divide each term in $4 a = - 16$ by $4$ and simplify.

Divide each term in $4 a = - 16$ by $4$ .

$\frac{4 a}{4} = \frac{- 16}{4}$

$6 = 2 a + 14$

$b = 14$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 a}{4} = \frac{- 16}{4}$

$6 = 2 a + 14$

$b = 14$

Divide $a$ by $1$ .

$a = \frac{- 16}{4}$

$6 = 2 a + 14$

$b = 14$

$a = \frac{- 16}{4}$

$6 = 2 a + 14$

$b = 14$

$a = \frac{- 16}{4}$

$6 = 2 a + 14$

$b = 14$

Simplify the right side.

Divide $- 16$ by $4$ .

$a = - 4$

$6 = 2 a + 14$

$b = 14$

$a = - 4$

$6 = 2 a + 14$

$b = 14$

$a = - 4$

$6 = 2 a + 14$

$b = 14$

$a = - 4$

$6 = 2 a + 14$

$b = 14$

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

Replace all occurrences of $a$ in $6 = 2 a + 14$ with $- 4$ .

$6 = 2 (- 4) + 14$

$a = - 4$

$b = 14$

Simplify the right side.

Simplify $2 (- 4) + 14$ .

Multiply $2$ by $- 4$ .

$6 = - 8 + 14$

$a = - 4$

$b = 14$

Add $- 8$ and $14$ .

$6 = 6$

$a = - 4$

$b = 14$

$6 = 6$

$a = - 4$

$b = 14$

$6 = 6$

$a = - 4$

$b = 14$

$6 = 6$

$a = - 4$

$b = 14$

Remove any equations from the system that are always true.

$a = - 4$

$b = 14$

List all of the solutions.

$a = - 4, b = 14$

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 $a = - 4$ , $b = 14$ , and $x = 0$ .

Multiply $- 4$ by $0$ .

$y = 0 + 14$

Add $0$ and $14$ .

$y = 14$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 0$ . This check passes since $y = 14$ and $y = 14$ .

$14 = 14$

Calculate the value of $y$ when $a = - 4$ , $b = 14$ , and $x = 2$ .

Multiply $- 4$ by $2$ .

$y = - 8 + 14$

Add $- 8$ and $14$ .

$y = 6$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 6$ and $y = 6$ .

$6 = 6$

Calculate the value of $y$ when $a = - 4$ , $b = 14$ , and $x = 4$ .

Multiply $- 4$ by $4$ .

$y = - 16 + 14$

Add $- 16$ and $14$ .

$y = - 2$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 4$ . This check passes since $y = - 2$ and $y = - 2$ .

$- 2 = - 2$

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 $y = - 4 x + 14$ .

$y = - 4 x + 14$

Find the Function Rule table[[x,y],[0,14],[2,6],[4,-2]]

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