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

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

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 $y = a x + b$ .

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

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

Rewrite the equation as $b = 4$ .

$b = 4$

$2 = a + b$

$1 = a (2) + b$

Replace all occurrences of $b$ with $4$ in each equation.

Replace all occurrences of $b$ in $2 = a + b$ with $4$ .

$2 = a + 4$

$b = 4$

$1 = a (2) + b$

Simplify the left side.

Remove parentheses.

$2 = a + 4$

$b = 4$

$1 = a (2) + b$

$2 = a + 4$

$b = 4$

$1 = a (2) + b$

Replace all occurrences of $b$ in $1 = a (2) + b$ with $4$ .

$1 = a (2) + 4$

$2 = a + 4$

$b = 4$

Simplify $1 = a (2) + 4$ .

Simplify the left side.

Remove parentheses.

$1 = a (2) + 4$

$2 = a + 4$

$b = 4$

$1 = a (2) + 4$

$2 = a + 4$

$b = 4$

Simplify the right side.

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

$1 = 2 a + 4$

$2 = a + 4$

$b = 4$

$1 = 2 a + 4$

$2 = a + 4$

$b = 4$

$1 = 2 a + 4$

$2 = a + 4$

$b = 4$

$1 = 2 a + 4$

$2 = a + 4$

$b = 4$

Solve for $a$ in $1 = 2 a + 4$ .

Rewrite the equation as $2 a + 4 = 1$ .

$2 a + 4 = 1$

$2 = a + 4$

$b = 4$

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

Subtract $4$ from both sides of the equation.

$2 a = 1 - 4$

$2 = a + 4$

$b = 4$

Subtract $4$ from $1$ .

$2 a = - 3$

$2 = a + 4$

$b = 4$

$2 a = - 3$

$2 = a + 4$

$b = 4$

Divide each term in $2 a = - 3$ by $2$ and simplify.

Divide each term in $2 a = - 3$ by $2$ .

$\frac{2 a}{2} = \frac{- 3}{2}$

$2 = a + 4$

$b = 4$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{- 3}{2}$

$2 = a + 4$

$b = 4$

Divide $a$ by $1$ .

$a = \frac{- 3}{2}$

$2 = a + 4$

$b = 4$

$a = \frac{- 3}{2}$

$2 = a + 4$

$b = 4$

$a = \frac{- 3}{2}$

$2 = a + 4$

$b = 4$

Simplify the right side.

Move the negative in front of the fraction.

$a = - \frac{3}{2}$

$2 = a + 4$

$b = 4$

$a = - \frac{3}{2}$

$2 = a + 4$

$b = 4$

$a = - \frac{3}{2}$

$2 = a + 4$

$b = 4$

$a = - \frac{3}{2}$

$2 = a + 4$

$b = 4$

Replace all occurrences of $a$ with $- \frac{3}{2}$ in each equation.

Replace all occurrences of $a$ in $2 = a + 4$ with $- \frac{3}{2}$ .

$2 = (- \frac{3}{2}) + 4$

$a = - \frac{3}{2}$

$b = 4$

Simplify the right side.

Simplify $(- \frac{3}{2}) + 4$ .

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 = - \frac{3}{2} + 4 \cdot \frac{2}{2}$

$a = - \frac{3}{2}$

$b = 4$

Combine $4$ and $\frac{2}{2}$ .

$2 = - \frac{3}{2} + \frac{4 \cdot 2}{2}$

$a = - \frac{3}{2}$

$b = 4$

Combine the numerators over the common denominator.

$2 = \frac{- 3 + 4 \cdot 2}{2}$

$a = - \frac{3}{2}$

$b = 4$

Simplify the numerator.

Multiply $4$ by $2$ .

$2 = \frac{- 3 + 8}{2}$

$a = - \frac{3}{2}$

$b = 4$

Add $- 3$ and $8$ .

$2 = \frac{5}{2}$

$a = - \frac{3}{2}$

$b = 4$

$2 = \frac{5}{2}$

$a = - \frac{3}{2}$

$b = 4$

$2 = \frac{5}{2}$

$a = - \frac{3}{2}$

$b = 4$

$2 = \frac{5}{2}$

$a = - \frac{3}{2}$

$b = 4$

$2 = \frac{5}{2}$

$a = - \frac{3}{2}$

$b = 4$

Since $2 = \frac{5}{2}$ is not true, there is no solution.

No solution

Since $y \neq y$ for the corresponding $x$ values, the function is not linear.

The function is not linear

Step 2

Check if the function rule is quadratic.

To find if the table follows a function rule, check whether the function rule could follow the form $y = a x^{2} + b x + c$ .

$y = a x^{2} + b x + c$

Build a set of $3$ equations from the table such that $y = a x^{2} + b x + c$ .

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

Solve for $c$ in $4 = a \cdot 0^{2} + c$ .

Rewrite the equation as $a \cdot 0^{2} + c = 4$ .

$a \cdot 0^{2} + c = 4$

$2 = a + b + c$