Find the Function Rule table[[x,y],[-2,-4.6],[-1,-2.6],[0,-0.6],[1,1.4],[2,3.4]]

$\begin{array}{c} x & y \\ - 2 & - 4.6 \\ - 1 & - 2.6 \\ 0 & - 0.6 \\ 1 & 1.4 \\ 2 & 3.4 \end{array}$

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 $y = a x + b$ .

$- 4.6 = a (- 2) + b - 2.6 = a (- 1) + b - 0.6 = a (0) + b 1.4 = a (1) + b 3.4 = a (2) + b$

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

Rewrite the equation as $b = - 0.6$ .

$b = - 0.6$

$- 4.6 = a (- 2) + b$

$- 2.6 = a (- 1) + b$

$1.4 = a + b$

$3.4 = a (2) + b$

Replace all occurrences of $b$ with $- 0.6$ in each equation.

Replace all occurrences of $b$ in $- 4.6 = a (- 2) + b$ with $- 0.6$ .

$- 4.6 = a (- 2) - 0.6$

$b = - 0.6$

$- 2.6 = a (- 1) + b$

$1.4 = a + b$

$3.4 = a (2) + b$

Simplify $- 4.6 = a (- 2) - 0.6$ .

Simplify the left side.

Remove parentheses.

$- 4.6 = a (- 2) - 0.6$

$b = - 0.6$

$- 2.6 = a (- 1) + b$

$1.4 = a + b$

$3.4 = a (2) + b$

$- 4.6 = a (- 2) - 0.6$

$b = - 0.6$

$- 2.6 = a (- 1) + b$

$1.4 = a + b$

$3.4 = a (2) + b$

Simplify the right side.

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

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$- 2.6 = a (- 1) + b$

$1.4 = a + b$

$3.4 = a (2) + b$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$- 2.6 = a (- 1) + b$

$1.4 = a + b$

$3.4 = a (2) + b$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$- 2.6 = a (- 1) + b$

$1.4 = a + b$

$3.4 = a (2) + b$

Replace all occurrences of $b$ in $- 2.6 = a (- 1) + b$ with $- 0.6$ .

$- 2.6 = a (- 1) - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$1.4 = a + b$

$3.4 = a (2) + b$

Simplify $- 2.6 = a (- 1) - 0.6$ .

Simplify the left side.

Remove parentheses.

$- 2.6 = a (- 1) - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$1.4 = a + b$

$3.4 = a (2) + b$

$- 2.6 = a (- 1) - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$1.4 = a + b$

$3.4 = a (2) + b$

Simplify the right side.

Simplify each term.

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

$- 2.6 = - 1 a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$1.4 = a + b$

$3.4 = a (2) + b$

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

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$1.4 = a + b$

$3.4 = a (2) + b$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$1.4 = a + b$

$3.4 = a (2) + b$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$1.4 = a + b$

$3.4 = a (2) + b$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$1.4 = a + b$

$3.4 = a (2) + b$

Replace all occurrences of $b$ in $1.4 = a + b$ with $- 0.6$ .

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$3.4 = a (2) + b$

Simplify the left side.

Remove parentheses.

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$3.4 = a (2) + b$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$3.4 = a (2) + b$

Replace all occurrences of $b$ in $3.4 = a (2) + b$ with $- 0.6$ .

$3.4 = a (2) - 0.6$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

Simplify $3.4 = a (2) - 0.6$ .

Simplify the left side.

Remove parentheses.

$3.4 = a (2) - 0.6$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$3.4 = a (2) - 0.6$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

Simplify the right side.

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

$3.4 = 2 a - 0.6$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$3.4 = 2 a - 0.6$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$3.4 = 2 a - 0.6$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$3.4 = 2 a - 0.6$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

Solve for $a$ in $3.4 = 2 a - 0.6$ .

Rewrite the equation as $2 a - 0.6 = 3.4$ .

$2 a - 0.6 = 3.4$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

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

Add $0.6$ to both sides of the equation.

$2 a = 3.4 + 0.6$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

Add $3.4$ and $0.6$ .

$2 a = 4$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$2 a = 4$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

Divide each term in $2 a = 4$ by $2$ and simplify.

Divide each term in $2 a = 4$ by $2$ .

$\frac{2 a}{2} = \frac{4}{2}$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{4}{2}$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

Divide $a$ by $1$ .

$a = \frac{4}{2}$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$a = \frac{4}{2}$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$a = \frac{4}{2}$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

Simplify the right side.

Divide $4$ by $2$ .

$a = 2$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$a = 2$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$a = 2$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$a = 2$

$1.4 = a - 0.6$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

Replace all occurrences of $a$ with $2$ in each equation.

Replace all occurrences of $a$ in $1.4 = a - 0.6$ with $2$ .

$1.4 = (2) - 0.6$

$a = 2$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

Simplify the right side.

Subtract $0.6$ from $2$ .

$1.4 = 1.4$

$a = 2$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$1.4 = 1.4$

$a = 2$

$- 2.6 = - a - 0.6$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

Replace all occurrences of $a$ in $- 2.6 = - a - 0.6$ with $2$ .

$- 2.6 = - (2) - 0.6$

$1.4 = 1.4$

$a = 2$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

Simplify the right side.

Simplify $- (2) - 0.6$ .

Multiply $- 1$ by $2$ .

$- 2.6 = - 2 - 0.6$

$1.4 = 1.4$

$a = 2$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

Subtract $0.6$ from $- 2$ .

$- 2.6 = - 2.6$

$1.4 = 1.4$

$a = 2$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$- 2.6 = - 2.6$

$1.4 = 1.4$

$a = 2$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

$- 2.6 = - 2.6$

$1.4 = 1.4$

$a = 2$

$- 4.6 = - 2 a - 0.6$

$b = - 0.6$

Replace all occurrences of $a$ in $- 4.6 = - 2 a - 0.6$ with $2$ .

$- 4.6 = - 2 \cdot 2 - 0.6$

$- 2.6 = - 2.6$

$1.4 = 1.4$

$a = 2$

$b = - 0.6$

Simplify the right side.

Simplify $- 2 \cdot 2 - 0.6$ .

Multiply $- 2$ by $2$ .

$- 4.6 = - 4 - 0.6$

$- 2.6 = - 2.6$

$1.4 = 1.4$

$a = 2$

$b = - 0.6$

Subtract $0.6$ from $- 4$ .

$- 4.6 = - 4.6$

$- 2.6 = - 2.6$

$1.4 = 1.4$

$a = 2$

$b = - 0.6$

$- 4.6 = - 4.6$

$- 2.6 = - 2.6$

$1.4 = 1.4$

$a = 2$

$b = - 0.6$

$- 4.6 = - 4.6$

$- 2.6 = - 2.6$

$1.4 = 1.4$

$a = 2$

$b = - 0.6$

$- 4.6 = - 4.6$

$- 2.6 = - 2.6$

$1.4 = 1.4$

$a = 2$

$b = - 0.6$

Remove any equations from the system that are always true.

$a = 2$

$b = - 0.6$

List all of the solutions.

$a = 2, b = - 0.6$

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 $a = 2$ , $b = - 0.6$ , and $x = - 2$ .

Multiply $2$ by $- 2$ .

$y = - 4 - 0.6$

Subtract $0.6$ from $- 4$ .

$y = - 4.6$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 2$ . This check passes since $y = - 4.6$ and $y = - 4.6$ .

$- 4.6 = - 4.6$

Calculate the value of $y$ when $a = 2$ , $b = - 0.6$ , and $x = - 1$ .

Multiply $2$ by $- 1$ .

$y = - 2 - 0.6$

Subtract $0.6$ from $- 2$ .

$y = - 2.6$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 1$ . This check passes since $y = - 2.6$ and $y = - 2.6$ .

$- 2.6 = - 2.6$

Calculate the value of $y$ when $a = 2$ , $b = - 0.6$ , and $x = 0$ .

Multiply $2$ by $0$ .

$y = 0 - 0.6$

Subtract $0.6$ from $0$ .

$y = - 0.6$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 0$ . This check passes since $y = - 0.6$ and $y = - 0.6$ .

$- 0.6 = - 0.6$

Calculate the value of $y$ when $a = 2$ , $b = - 0.6$ , and $x = 1$ .

Multiply $2$ by $1$ .

$y = 2 - 0.6$

Subtract $0.6$ from $2$ .

$y = 1.4$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = 1.4$ and $y = 1.4$ .

$1.4 = 1.4$

Calculate the value of $y$ when $a = 2$ , $b = - 0.6$ , and $x = 2$ .

Multiply $2$ by $2$ .

$y = 4 - 0.6$

Subtract $0.6$ from $4$ .

$y = 3.4$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 3.4$ and $y = 3.4$ .

$3.4 = 3.4$

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 $y = 2 x - 0.6$ .

$y = 2 x - 0.6$

Find the Function Rule table[[x,y],[-2,-4.6],[-1,-2.6],[0,-0.6],[1,1.4],[2,3.4]]

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