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

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

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 $y = a x + b$ .

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

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

Rewrite the equation as $b = 1$ .

$b = 1$

$0 = a (- 1) + b$

$2 = a + b$

Replace all occurrences of $b$ with $1$ in each equation.

Replace all occurrences of $b$ in $0 = a (- 1) + b$ with $1$ .

$0 = a (- 1) + 1$

$b = 1$

$2 = a + b$

Simplify $0 = a (- 1) + 1$ .

Simplify the left side.

Remove parentheses.

$0 = a (- 1) + 1$

$b = 1$

$2 = a + b$

$0 = a (- 1) + 1$

$b = 1$

$2 = a + b$

Simplify the right side.

Simplify each term.

Move $- 1$ to the left of $a$ .

$0 = - 1 a + 1$

$b = 1$

$2 = a + b$

Rewrite $- 1 a$ as $- a$ .

$0 = - a + 1$

$b = 1$

$2 = a + b$

$0 = - a + 1$

$b = 1$

$2 = a + b$

$0 = - a + 1$

$b = 1$

$2 = a + b$

$0 = - a + 1$

$b = 1$

$2 = a + b$

Replace all occurrences of $b$ in $2 = a + b$ with $1$ .

$2 = a + 1$

$0 = - a + 1$

$b = 1$

Simplify the left side.

Remove parentheses.

$2 = a + 1$

$0 = - a + 1$

$b = 1$

$2 = a + 1$

$0 = - a + 1$

$b = 1$

$2 = a + 1$

$0 = - a + 1$

$b = 1$

Solve for $a$ in $2 = a + 1$ .

Rewrite the equation as $a + 1 = 2$ .

$a + 1 = 2$

$0 = - a + 1$

$b = 1$

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

Subtract $1$ from both sides of the equation.

$a = 2 - 1$

$0 = - a + 1$

$b = 1$

Subtract $1$ from $2$ .

$a = 1$

$0 = - a + 1$

$b = 1$

$a = 1$

$0 = - a + 1$

$b = 1$

$a = 1$

$0 = - a + 1$

$b = 1$

Replace all occurrences of $a$ with $1$ in each equation.

Replace all occurrences of $a$ in $0 = - a + 1$ with $1$ .

$0 = - (1) + 1$

$a = 1$

$b = 1$

Simplify the right side.

Simplify $- (1) + 1$ .

Multiply $- 1$ by $1$ .

$0 = - 1 + 1$

$a = 1$

$b = 1$

Add $- 1$ and $1$ .

$0 = 0$

$a = 1$

$b = 1$

$0 = 0$

$a = 1$

$b = 1$

$0 = 0$

$a = 1$

$b = 1$

$0 = 0$

$a = 1$

$b = 1$

Remove any equations from the system that are always true.

$a = 1$

$b = 1$

List all of the solutions.

$a = 1, b = 1$

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 $a = 1$ , $b = 1$ , and $x = - 1$ .

Multiply $- 1$ by $1$ .

$y = - 1 + 1$

Add $- 1$ and $1$ .

$y = 0$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 1$ . This check passes since $y = 0$ and $y = 0$ .

$0 = 0$

Calculate the value of $y$ when $a = 1$ , $b = 1$ , and $x = 0$ .

Multiply $0$ by $1$ .

$y = 0 + 1$

Add $0$ and $1$ .

$y = 1$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 0$ . This check passes since $y = 1$ and $y = 1$ .

$1 = 1$

Calculate the value of $y$ when $a = 1$ , $b = 1$ , and $x = 1$ .

Multiply $1$ by $1$ .

$y = 1 + 1$

Add $1$ and $1$ .

$y = 2$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . 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 $y = x + 1$ .

$y = x + 1$

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

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