Find the Function Rule table[[x,y],[0,23000],[1,20000],[2,16000]]

$\begin{array}{c} x & y \\ 0 & 23000 \\ 1 & 20000 \\ 2 & 16000 \end{array}$

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 $y = a x + b$ .

$23000 = a (0) + b 20000 = a (1) + b 16000 = a (2) + b$

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

Rewrite the equation as $b = 23000$ .

$b = 23000$

$20000 = a + b$

$16000 = a (2) + b$

Replace all occurrences of $b$ with $23000$ in each equation.

Replace all occurrences of $b$ in $20000 = a + b$ with $23000$ .

$20000 = a + 23000$

$b = 23000$

$16000 = a (2) + b$

Simplify the left side.

Remove parentheses.

$20000 = a + 23000$

$b = 23000$

$16000 = a (2) + b$

$20000 = a + 23000$

$b = 23000$

$16000 = a (2) + b$

Replace all occurrences of $b$ in $16000 = a (2) + b$ with $23000$ .

$16000 = a (2) + 23000$

$20000 = a + 23000$

$b = 23000$

Simplify $16000 = a (2) + 23000$ .

Simplify the left side.

Remove parentheses.

$16000 = a (2) + 23000$

$20000 = a + 23000$

$b = 23000$

$16000 = a (2) + 23000$

$20000 = a + 23000$

$b = 23000$

Simplify the right side.

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

$16000 = 2 a + 23000$

$20000 = a + 23000$

$b = 23000$

$16000 = 2 a + 23000$

$20000 = a + 23000$

$b = 23000$

$16000 = 2 a + 23000$

$20000 = a + 23000$

$b = 23000$

$16000 = 2 a + 23000$

$20000 = a + 23000$

$b = 23000$

Solve for $a$ in $16000 = 2 a + 23000$ .

Rewrite the equation as $2 a + 23000 = 16000$ .

$2 a + 23000 = 16000$

$20000 = a + 23000$

$b = 23000$

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

Subtract $23000$ from both sides of the equation.

$2 a = 16000 - 23000$

$20000 = a + 23000$

$b = 23000$

Subtract $23000$ from $16000$ .

$2 a = - 7000$

$20000 = a + 23000$

$b = 23000$

$2 a = - 7000$

$20000 = a + 23000$

$b = 23000$

Divide each term in $2 a = - 7000$ by $2$ and simplify.

Divide each term in $2 a = - 7000$ by $2$ .

$\frac{2 a}{2} = \frac{- 7000}{2}$

$20000 = a + 23000$

$b = 23000$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{- 7000}{2}$

$20000 = a + 23000$

$b = 23000$

Divide $a$ by $1$ .

$a = \frac{- 7000}{2}$

$20000 = a + 23000$

$b = 23000$

$a = \frac{- 7000}{2}$

$20000 = a + 23000$

$b = 23000$

$a = \frac{- 7000}{2}$

$20000 = a + 23000$

$b = 23000$

Simplify the right side.

Divide $- 7000$ by $2$ .

$a = - 3500$

$20000 = a + 23000$

$b = 23000$

$a = - 3500$

$20000 = a + 23000$

$b = 23000$

$a = - 3500$

$20000 = a + 23000$

$b = 23000$

$a = - 3500$

$20000 = a + 23000$

$b = 23000$

Replace all occurrences of $a$ with $- 3500$ in each equation.

Replace all occurrences of $a$ in $20000 = a + 23000$ with $- 3500$ .

$20000 = (- 3500) + 23000$

$a = - 3500$

$b = 23000$

Simplify the right side.

Add $- 3500$ and $23000$ .

$20000 = 19500$

$a = - 3500$

$b = 23000$

$20000 = 19500$

$a = - 3500$

$b = 23000$

$20000 = 19500$

$a = - 3500$

$b = 23000$

Since $20000 = 19500$ 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

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 $23000 = a \cdot 0^{2} + c$ .

Rewrite the equation as $a \cdot 0^{2} + c = 23000$ .

$a \cdot 0^{2} + c = 23000$

$20000 = a + b + c$