Find the Function Rule table[[x,y],[0,0],[1,4],[2,8]]

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 4 \\ 2 & 8 \end{array}$

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 $y = a x + b$ .

$0 = a (0) + b 4 = a (1) + b 8 = a (2) + b$

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

Rewrite the equation as $b = 0$ .

$b = 0$

$4 = a + b$

$8 = a (2) + b$

Replace all occurrences of $b$ with $0$ in each equation.

Replace all occurrences of $b$ in $4 = a + b$ with $0$ .

$4 = a + 0$

$b = 0$

$8 = a (2) + b$

Simplify $4 = a + 0$ .

Simplify the left side.

Remove parentheses.

$4 = a + 0$

$b = 0$

$8 = a (2) + b$

$4 = a + 0$

$b = 0$

$8 = a (2) + b$

Simplify the right side.

Add $a$ and $0$ .

$4 = a$

$b = 0$

$8 = a (2) + b$

$4 = a$

$b = 0$

$8 = a (2) + b$

$4 = a$

$b = 0$

$8 = a (2) + b$

Replace all occurrences of $b$ in $8 = a (2) + b$ with $0$ .

$8 = a (2) + 0$

$4 = a$

$b = 0$

Simplify $8 = a (2) + 0$ .

Simplify the left side.

Remove parentheses.

$8 = a (2) + 0$

$4 = a$

$b = 0$

$8 = a (2) + 0$

$4 = a$

$b = 0$

Simplify the right side.

Simplify $a (2) + 0$ .

Add $a (2)$ and $0$ .

$8 = a (2)$

$4 = a$

$b = 0$

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

$8 = 2 a$

$4 = a$

$b = 0$

$8 = 2 a$

$4 = a$

$b = 0$

$8 = 2 a$

$4 = a$

$b = 0$

$8 = 2 a$

$4 = a$

$b = 0$

$8 = 2 a$

$4 = a$

$b = 0$

Solve for $a$ in $8 = 2 a$ .

Rewrite the equation as $2 a = 8$ .

$2 a = 8$

$4 = a$

$b = 0$

Divide each term in $2 a = 8$ by $2$ and simplify.

Divide each term in $2 a = 8$ by $2$ .

$\frac{2 a}{2} = \frac{8}{2}$

$4 = a$

$b = 0$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{8}{2}$

$4 = a$

$b = 0$

Divide $a$ by $1$ .

$a = \frac{8}{2}$

$4 = a$

$b = 0$

$a = \frac{8}{2}$

$4 = a$

$b = 0$

$a = \frac{8}{2}$

$4 = a$

$b = 0$

Simplify the right side.

Divide $8$ by $2$ .

$a = 4$

$4 = a$

$b = 0$

$a = 4$

$4 = a$

$b = 0$

$a = 4$

$4 = a$

$b = 0$

$a = 4$

$4 = a$

$b = 0$

List all of the solutions.

$a = 4, b = 0$

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

Multiply $4$ by $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

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

$0 = 0$

Calculate the value of $y$ when $a = 4$ , $b = 0$ , and $x = 1$ .

Multiply $4$ by $1$ .

$y = 4 + 0$

Add $4$ and $0$ .

$y = 4$

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 $a = 4$ , $b = 0$ , and $x = 2$ .

Multiply $4$ by $2$ .

$y = 8 + 0$

Add $8$ and $0$ .

$y = 8$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 8$ and $y = 8$ .

$8 = 8$

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 $y = 4 x$ .

$y = 4 x$

Find the Function Rule table[[x,y],[0,0],[1,4],[2,8]]

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