Find the Function Rule table[[x,y],[-2,12],[0,3],[3,-10.5]]

$\begin{array}{c} x & y \\ - 2 & 12 \\ 0 & 3 \\ 3 & - 10.5 \end{array}$

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 $y = a x + b$ .

$12 = a (- 2) + b 3 = a (0) + b - 10.5 = a (3) + b$

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

Rewrite the equation as $b = 3$ .

$b = 3$

$12 = a (- 2) + b$

$- 10.5 = a (3) + b$

Replace all occurrences of $b$ with $3$ in each equation.

Replace all occurrences of $b$ in $12 = a (- 2) + b$ with $3$ .

$12 = a (- 2) + 3$

$b = 3$

$- 10.5 = a (3) + b$

Simplify $12 = a (- 2) + 3$ .

Simplify the left side.

Remove parentheses.

$12 = a (- 2) + 3$

$b = 3$

$- 10.5 = a (3) + b$

$12 = a (- 2) + 3$

$b = 3$

$- 10.5 = a (3) + b$

Simplify the right side.

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

$12 = - 2 a + 3$

$b = 3$

$- 10.5 = a (3) + b$

$12 = - 2 a + 3$

$b = 3$

$- 10.5 = a (3) + b$

$12 = - 2 a + 3$

$b = 3$

$- 10.5 = a (3) + b$

Replace all occurrences of $b$ in $- 10.5 = a (3) + b$ with $3$ .

$- 10.5 = a (3) + 3$

$12 = - 2 a + 3$

$b = 3$

Simplify $- 10.5 = a (3) + 3$ .

Simplify the left side.

Remove parentheses.

$- 10.5 = a (3) + 3$

$12 = - 2 a + 3$

$b = 3$

$- 10.5 = a (3) + 3$

$12 = - 2 a + 3$

$b = 3$

Simplify the right side.

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

$- 10.5 = 3 a + 3$

$12 = - 2 a + 3$

$b = 3$

$- 10.5 = 3 a + 3$

$12 = - 2 a + 3$

$b = 3$

$- 10.5 = 3 a + 3$

$12 = - 2 a + 3$

$b = 3$

$- 10.5 = 3 a + 3$

$12 = - 2 a + 3$

$b = 3$

Solve for $a$ in $- 10.5 = 3 a + 3$ .

Rewrite the equation as $3 a + 3 = - 10.5$ .

$3 a + 3 = - 10.5$

$12 = - 2 a + 3$

$b = 3$

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

Subtract $3$ from both sides of the equation.

$3 a = - 10.5 - 3$

$12 = - 2 a + 3$

$b = 3$

Subtract $3$ from $- 10.5$ .

$3 a = - 13.5$

$12 = - 2 a + 3$

$b = 3$

$3 a = - 13.5$

$12 = - 2 a + 3$

$b = 3$

Divide each term in $3 a = - 13.5$ by $3$ and simplify.

Divide each term in $3 a = - 13.5$ by $3$ .

$\frac{3 a}{3} = \frac{- 13.5}{3}$

$12 = - 2 a + 3$

$b = 3$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 a}{3} = \frac{- 13.5}{3}$

$12 = - 2 a + 3$

$b = 3$

Divide $a$ by $1$ .

$a = \frac{- 13.5}{3}$

$12 = - 2 a + 3$

$b = 3$

$a = \frac{- 13.5}{3}$

$12 = - 2 a + 3$

$b = 3$

$a = \frac{- 13.5}{3}$

$12 = - 2 a + 3$

$b = 3$

Simplify the right side.

Divide $- 13.5$ by $3$ .

$a = - 4.5$

$12 = - 2 a + 3$

$b = 3$

$a = - 4.5$

$12 = - 2 a + 3$

$b = 3$

$a = - 4.5$

$12 = - 2 a + 3$

$b = 3$

$a = - 4.5$

$12 = - 2 a + 3$

$b = 3$

Replace all occurrences of $a$ with $- 4.5$ in each equation.

Replace all occurrences of $a$ in $12 = - 2 a + 3$ with $- 4.5$ .

$12 = - 2 \cdot - 4.5 + 3$

$a = - 4.5$

$b = 3$

Simplify the right side.

Simplify $- 2 \cdot - 4.5 + 3$ .

Multiply $- 2$ by $- 4.5$ .

$12 = 9 + 3$

$a = - 4.5$

$b = 3$

Add $9$ and $3$ .

$12 = 12$

$a = - 4.5$

$b = 3$

$12 = 12$

$a = - 4.5$

$b = 3$

$12 = 12$

$a = - 4.5$

$b = 3$

$12 = 12$

$a = - 4.5$

$b = 3$

Remove any equations from the system that are always true.

$a = - 4.5$

$b = 3$

List all of the solutions.

$a = - 4.5, b = 3$

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 $a = - 4.5$ , $b = 3$ , and $x = - 2$ .

Multiply $- 4.5$ by $- 2$ .

$y = 9 + 3$

Add $9$ and $3$ .

$y = 12$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 2$ . This check passes since $y = 12$ and $y = 12$ .

$12 = 12$

Calculate the value of $y$ when $a = - 4.5$ , $b = 3$ , and $x = 0$ .

Multiply $- 4.5$ by $0$ .

$y = 0 + 3$

Add $0$ and $3$ .

$y = 3$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 0$ . This check passes since $y = 3$ and $y = 3$ .

$3 = 3$

Calculate the value of $y$ when $a = - 4.5$ , $b = 3$ , and $x = 3$ .

Multiply $- 4.5$ by $3$ .

$y = - 13.5 + 3$

Add $- 13.5$ and $3$ .

$y = - 10.5$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = - 10.5$ and $y = - 10.5$ .

$- 10.5 = - 10.5$

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 $y = - 4.5 x + 3$ .

$y = - 4.5 x + 3$

Find the Function Rule table[[x,y],[-2,12],[0,3],[3,-10.5]]

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