Find the Function Rule table[[x,y],[-3,4],[0,-3],[1,2]]

$\begin{array}{c} x & y \\ - 3 & 4 \\ 0 & - 3 \\ 1 & 2 \end{array}$

Check if the function rule is linear.

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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 $y = a x + b$ .

$4 = a (- 3) + b - 3 = a (0) + b 2 = a (1) + b$

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

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Simplify each equation.

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Simplify $a (0) + b$ .

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Multiply $a$ by $0$ .

$- 3 = 0 + b$

$4 = a (- 3) + b$

$2 = a (1) + b$

Add $0$ and $b$ .

$- 3 = b$

$4 = a (- 3) + b$

$2 = a (1) + b$

$- 3 = b$

$4 = a (- 3) + b$

$2 = a (1) + b$

Move $- 3$ to the left of $a$ .

$- 3 = b$

$4 = - 3 a + b$

$2 = a (1) + b$

Multiply $a$ by $1$ .

$- 3 = b$

$4 = - 3 a + b$

$2 = a + b$

$- 3 = b$

$4 = - 3 a + b$

$2 = a + b$

Rewrite the equation as $b = - 3$ .

$b = - 3$

$4 = - 3 a + b$

$2 = a + b$

Replace all occurrences of $b$ with $- 3$ in each equation.

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Replace all occurrences of $b$ in $4 = - 3 a + b$ with $- 3$ .

$b = - 3$

$4 = - 3 a - 3$

$2 = a + b$

Replace all occurrences of $b$ in $2 = a + b$ with $- 3$ .

$b = - 3$

$4 = - 3 a - 3$

$2 = a - 3$

$b = - 3$

$4 = - 3 a - 3$

$2 = a - 3$

Simplify.

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Remove parentheses.

$b = - 3$

$4 = - 3 a - 3$

$2 = a - 3$

Remove parentheses.

$b = - 3$

$4 = - 3 a - 3$

$2 = a - 3$

$b = - 3$

$4 = - 3 a - 3$

$2 = a - 3$

Solve for $a$ in the third equation.

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Rewrite the equation as $a - 3 = 2$ .

$b = - 3$

$4 = - 3 a - 3$

$a - 3 = 2$

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

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Add $3$ to both sides of the equation.

$b = - 3$

$4 = - 3 a - 3$

$a = 2 + 3$

Add $2$ and $3$ .

$b = - 3$

$4 = - 3 a - 3$

$a = 5$

$b = - 3$

$4 = - 3 a - 3$

$a = 5$

$b = - 3$

$4 = - 3 a - 3$

$a = 5$

Replace all occurrences of $a$ in $4 = - 3 a - 3$ with $5$ .

$b = - 3$

$4 = - 3 \cdot 5 - 3$

$a = 5$

Simplify $- 3 \cdot 5 - 3$ .

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Multiply $- 3$ by $5$ .

$b = - 3$

$4 = - 15 - 3$

$a = 5$

Subtract $3$ from $- 15$ .

$b = - 3$

$4 = - 18$

$a = 5$

$b = - 3$

$4 = - 18$

$a = 5$

Since $4 \neq - 18$ , there are no solutions.

$b = - 3$

No solution

$a = 5$

$b = - 3$

No solution

$a = 5$

Calculate the value of $y$ using each $x$ value in the relation and compare this value to the given $y$ value in the relation.

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Calculate the value of $y$ when $a = 5$ , $b = - 3$ , and $x = - 3$ .

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Multiply $5$ by $- 3$ .

$y = - 15 - 3$

Subtract $3$ from $- 15$ .

$y = - 18$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 3$ . This check does not pass, since $y = - 18$ and $y = 4$ . The function rule can’t be linear.

$- 18 \neq 4$

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.

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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 $c$ .

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Simplify each equation.

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Simplify $a {(0)}^{2} + b (0) + c$ .

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Simplify each term.

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Raising $0$ to any positive power yields $0$ .

$- 3 = a \cdot 0 + b (0) + c$