Find the Function Rule table[[x,y],[-1,1],[0,3],[1,9],[2,27],[3,81]]

$\begin{array}{c} x & y \\ - 1 & 1 \\ 0 & 3 \\ 1 & 9 \\ 2 & 27 \\ 3 & 81 \end{array}$

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 $y = a x + b$ .

$1 = a (- 1) + b 3 = a (0) + b 9 = a (1) + b 27 = a (2) + b 81 = a (3) + b$

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

Rewrite the equation as $b = 3$ .

$b = 3$

$1 = a (- 1) + b$

$9 = a + b$

$27 = a (2) + b$

$81 = a (3) + b$

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

Replace all occurrences of $b$ in $1 = a (- 1) + b$ with $3$ .

$1 = a (- 1) + 3$

$b = 3$

$9 = a + b$

$27 = a (2) + b$

$81 = a (3) + b$

Simplify $1 = a (- 1) + 3$ .

Simplify the left side.

Remove parentheses.

$1 = a (- 1) + 3$

$b = 3$

$9 = a + b$

$27 = a (2) + b$

$81 = a (3) + b$

$1 = a (- 1) + 3$

$b = 3$

$9 = a + b$

$27 = a (2) + b$

$81 = a (3) + b$

Simplify the right side.

Simplify each term.

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

$1 = - 1 a + 3$

$b = 3$

$9 = a + b$

$27 = a (2) + b$

$81 = a (3) + b$

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

$1 = - a + 3$

$b = 3$

$9 = a + b$

$27 = a (2) + b$

$81 = a (3) + b$

$1 = - a + 3$

$b = 3$

$9 = a + b$

$27 = a (2) + b$

$81 = a (3) + b$

$1 = - a + 3$

$b = 3$

$9 = a + b$

$27 = a (2) + b$

$81 = a (3) + b$

$1 = - a + 3$

$b = 3$

$9 = a + b$

$27 = a (2) + b$

$81 = a (3) + b$

Replace all occurrences of $b$ in $9 = a + b$ with $3$ .

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$27 = a (2) + b$

$81 = a (3) + b$

Simplify the left side.

Remove parentheses.

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$27 = a (2) + b$

$81 = a (3) + b$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$27 = a (2) + b$

$81 = a (3) + b$

Replace all occurrences of $b$ in $27 = a (2) + b$ with $3$ .

$27 = a (2) + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$81 = a (3) + b$

Simplify $27 = a (2) + 3$ .

Simplify the left side.

Remove parentheses.

$27 = a (2) + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$81 = a (3) + b$

$27 = a (2) + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$81 = a (3) + b$

Simplify the right side.

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

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$81 = a (3) + b$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$81 = a (3) + b$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$81 = a (3) + b$

Replace all occurrences of $b$ in $81 = a (3) + b$ with $3$ .

$81 = a (3) + 3$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

Simplify $81 = a (3) + 3$ .

Simplify the left side.

Remove parentheses.

$81 = a (3) + 3$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$81 = a (3) + 3$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

Simplify the right side.

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

$81 = 3 a + 3$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$81 = 3 a + 3$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$81 = 3 a + 3$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$81 = 3 a + 3$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

Solve for $a$ in $81 = 3 a + 3$ .

Rewrite the equation as $3 a + 3 = 81$ .

$3 a + 3 = 81$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

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

Subtract $3$ from both sides of the equation.

$3 a = 81 - 3$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

Subtract $3$ from $81$ .

$3 a = 78$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$3 a = 78$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

Divide each term in $3 a = 78$ by $3$ and simplify.

Divide each term in $3 a = 78$ by $3$ .

$\frac{3 a}{3} = \frac{78}{3}$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 a}{3} = \frac{78}{3}$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

Divide $a$ by $1$ .

$a = \frac{78}{3}$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$a = \frac{78}{3}$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$a = \frac{78}{3}$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

Simplify the right side.

Divide $78$ by $3$ .

$a = 26$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$a = 26$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$a = 26$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$a = 26$

$27 = 2 a + 3$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

Replace all occurrences of $a$ with $26$ in each equation.

Replace all occurrences of $a$ in $27 = 2 a + 3$ with $26$ .

$27 = 2 (26) + 3$

$a = 26$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

Simplify the right side.

Simplify $2 (26) + 3$ .

Multiply $2$ by $26$ .

$27 = 52 + 3$

$a = 26$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

Add $52$ and $3$ .

$27 = 55$

$a = 26$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$27 = 55$

$a = 26$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

$27 = 55$

$a = 26$

$9 = a + 3$

$1 = - a + 3$

$b = 3$

Replace all occurrences of $a$ in $9 = a + 3$ with $26$ .

$9 = (26) + 3$

$27 = 55$

$a = 26$

$1 = - a + 3$

$b = 3$

Simplify the right side.

Add $26$ and $3$ .

$9 = 29$

$27 = 55$

$a = 26$

$1 = - a + 3$

$b = 3$

$9 = 29$

$27 = 55$

$a = 26$

$1 = - a + 3$

$b = 3$

Replace all occurrences of $a$ in $1 = - a + 3$ with $26$ .

$1 = - (26) + 3$

$9 = 29$

$27 = 55$

$a = 26$

$b = 3$

Simplify the right side.

Simplify $- (26) + 3$ .

Multiply $- 1$ by $26$ .

$1 = - 26 + 3$

$9 = 29$

$27 = 55$

$a = 26$

$b = 3$

Add $- 26$ and $3$ .

$1 = - 23$

$9 = 29$

$27 = 55$

$a = 26$

$b = 3$

$1 = - 23$

$9 = 29$

$27 = 55$

$a = 26$

$b = 3$

$1 = - 23$

$9 = 29$

$27 = 55$

$a = 26$

$b = 3$

$1 = - 23$

$9 = 29$

$27 = 55$

$a = 26$

$b = 3$

Since $1 = - 23$ 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 $3 = a \cdot 0^{2} + c$ .

Rewrite the equation as $a \cdot 0^{2} + c = 3$ .

$a \cdot 0^{2} + c = 3$

$1 = a {(- 1)}^{2} + b (- 1) + c$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

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

Simplify each term.

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

$a \cdot 0 + c = 3$

$1 = a {(- 1)}^{2} + b (- 1) + c$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

Multiply $a$ by $0$ .

$0 + c = 3$

$1 = a {(- 1)}^{2} + b (- 1) + c$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

$0 + c = 3$

$1 = a {(- 1)}^{2} + b (- 1) + c$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

Add $0$ and $c$ .

$c = 3$

$1 = a {(- 1)}^{2} + b (- 1) + c$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

$c = 3$

$1 = a {(- 1)}^{2} + b (- 1) + c$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

$c = 3$

$1 = a {(- 1)}^{2} + b (- 1) + c$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

Replace all occurrences of $c$ with $3$ in each equation.

Replace all occurrences of $c$ in $1 = a {(- 1)}^{2} + b (- 1) + c$ with $3$ .

$1 = a {(- 1)}^{2} + b (- 1) + 3$

$c = 3$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

Simplify $1 = a {(- 1)}^{2} + b (- 1) + 3$ .

Simplify the left side.

Remove parentheses.

$1 = a {(- 1)}^{2} + b (- 1) + 3$

$c = 3$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

$1 = a {(- 1)}^{2} + b (- 1) + 3$

$c = 3$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

Simplify the right side.

Simplify each term.

Raise $- 1$ to the power of $2$ .

$1 = a \cdot 1 + b (- 1) + 3$

$c = 3$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

Multiply $a$ by $1$ .

$1 = a + b (- 1) + 3$

$c = 3$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

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

$1 = a - 1 b + 3$

$c = 3$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

Rewrite $- 1 b$ as $- b$ .

$1 = a - b + 3$

$c = 3$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

$1 = a - b + 3$

$c = 3$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

$1 = a - b + 3$

$c = 3$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

$1 = a - b + 3$

$c = 3$

$9 = a + b + c$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

Replace all occurrences of $c$ in $9 = a + b + c$ with $3$ .

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

Simplify the left side.

Remove parentheses.

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$27 = a \cdot 2^{2} + b (2) + c$

$81 = a \cdot 3^{2} + b (3) + c$

Replace all occurrences of $c$ in $27 = a \cdot 2^{2} + b (2) + c$ with $3$ .

$27 = a \cdot 2^{2} + b (2) + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$81 = a \cdot 3^{2} + b (3) + c$

Simplify $27 = a \cdot 2^{2} + b (2) + 3$ .

Simplify the left side.

Remove parentheses.

$27 = a \cdot 2^{2} + b (2) + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$81 = a \cdot 3^{2} + b (3) + c$

$27 = a \cdot 2^{2} + b (2) + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$81 = a \cdot 3^{2} + b (3) + c$

Simplify the right side.

Simplify each term.

Raise $2$ to the power of $2$ .

$27 = a \cdot 4 + b (2) + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$81 = a \cdot 3^{2} + b (3) + c$

Move $4$ to the left of $a$ .

$27 = 4 \cdot a + b (2) + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$81 = a \cdot 3^{2} + b (3) + c$

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

$27 = 4 a + 2 b + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$81 = a \cdot 3^{2} + b (3) + c$

$27 = 4 a + 2 b + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$81 = a \cdot 3^{2} + b (3) + c$

$27 = 4 a + 2 b + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$81 = a \cdot 3^{2} + b (3) + c$

$27 = 4 a + 2 b + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$81 = a \cdot 3^{2} + b (3) + c$

Replace all occurrences of $c$ in $81 = a \cdot 3^{2} + b (3) + c$ with $3$ .

$81 = a \cdot 3^{2} + b (3) + 3$

$27 = 4 a + 2 b + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

Simplify $81 = a \cdot 3^{2} + b (3) + 3$ .

Simplify the left side.

Remove parentheses.

$81 = a \cdot 3^{2} + b (3) + 3$

$27 = 4 a + 2 b + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$81 = a \cdot 3^{2} + b (3) + 3$

$27 = 4 a + 2 b + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

Simplify the right side.

Simplify each term.

Raise $3$ to the power of $2$ .

$81 = a \cdot 9 + b (3) + 3$

$27 = 4 a + 2 b + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

Move $9$ to the left of $a$ .

$81 = 9 \cdot a + b (3) + 3$

$27 = 4 a + 2 b + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

Move $3$ to the left of $b$ .

$81 = 9 a + 3 b + 3$

$27 = 4 a + 2 b + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$81 = 9 a + 3 b + 3$

$27 = 4 a + 2 b + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$81 = 9 a + 3 b + 3$

$27 = 4 a + 2 b + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$81 = 9 a + 3 b + 3$

$27 = 4 a + 2 b + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

$81 = 9 a + 3 b + 3$

$27 = 4 a + 2 b + 3$

$9 = a + b + 3$

$1 = a - b + 3$

$c = 3$

Solve for $a$ in $9 = a + b + 3$ .

Rewrite the equation as $a + b + 3 = 9$ .

$a + b + 3 = 9$

$81 = 9 a + 3 b + 3$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

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

Subtract $b$ from both sides of the equation.

$a + 3 = 9 - b$

$81 = 9 a + 3 b + 3$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

Subtract $3$ from both sides of the equation.

$a = 9 - b - 3$

$81 = 9 a + 3 b + 3$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

Subtract $3$ from $9$ .

$a = - b + 6$

$81 = 9 a + 3 b + 3$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

$a = - b + 6$

$81 = 9 a + 3 b + 3$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

$a = - b + 6$

$81 = 9 a + 3 b + 3$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

Replace all occurrences of $a$ with $- b + 6$ in each equation.

Replace all occurrences of $a$ in $81 = 9 a + 3 b + 3$ with $- b + 6$ .

$81 = 9 (- b + 6) + 3 b + 3$

$a = - b + 6$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

Simplify the right side.

Simplify $9 (- b + 6) + 3 b + 3$ .

Simplify each term.

Apply the distributive property.

$81 = 9 (- b) + 9 \cdot 6 + 3 b + 3$

$a = - b + 6$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

Multiply $- 1$ by $9$ .

$81 = - 9 b + 9 \cdot 6 + 3 b + 3$

$a = - b + 6$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

Multiply $9$ by $6$ .

$81 = - 9 b + 54 + 3 b + 3$

$a = - b + 6$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

$81 = - 9 b + 54 + 3 b + 3$

$a = - b + 6$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

Simplify by adding terms.

Add $- 9 b$ and $3 b$ .

$81 = - 6 b + 54 + 3$

$a = - b + 6$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

Add $54$ and $3$ .

$81 = - 6 b + 57$

$a = - b + 6$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

$81 = - 6 b + 57$

$a = - b + 6$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

$81 = - 6 b + 57$

$a = - b + 6$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

$81 = - 6 b + 57$

$a = - b + 6$

$27 = 4 a + 2 b + 3$

$1 = a - b + 3$

$c = 3$

Replace all occurrences of $a$ in $27 = 4 a + 2 b + 3$ with $- b + 6$ .

$27 = 4 (- b + 6) + 2 b + 3$

$81 = - 6 b + 57$

$a = - b + 6$

$1 = a - b + 3$

$c = 3$

Simplify the right side.

Simplify $4 (- b + 6) + 2 b + 3$ .

Simplify each term.

Apply the distributive property.

$27 = 4 (- b) + 4 \cdot 6 + 2 b + 3$

$81 = - 6 b + 57$

$a = - b + 6$

$1 = a - b + 3$

$c = 3$

Multiply $- 1$ by $4$ .

$27 = - 4 b + 4 \cdot 6 + 2 b + 3$

$81 = - 6 b + 57$

$a = - b + 6$

$1 = a - b + 3$

$c = 3$

Multiply $4$ by $6$ .

$27 = - 4 b + 24 + 2 b + 3$

$81 = - 6 b + 57$

$a = - b + 6$

$1 = a - b + 3$

$c = 3$

$27 = - 4 b + 24 + 2 b + 3$

$81 = - 6 b + 57$

$a = - b + 6$

$1 = a - b + 3$

$c = 3$

Simplify by adding terms.

Add $- 4 b$ and $2 b$ .

$27 = - 2 b + 24 + 3$

$81 = - 6 b + 57$

$a = - b + 6$

$1 = a - b + 3$

$c = 3$

Add $24$ and $3$ .

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$1 = a - b + 3$

$c = 3$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$1 = a - b + 3$

$c = 3$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$1 = a - b + 3$

$c = 3$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$1 = a - b + 3$

$c = 3$

Replace all occurrences of $a$ in $1 = a - b + 3$ with $- b + 6$ .

$1 = (- b + 6) - b + 3$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

Simplify the right side.

Simplify $(- b + 6) - b + 3$ .

Subtract $b$ from $- b$ .

$1 = - 2 b + 6 + 3$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

Add $6$ and $3$ .

$1 = - 2 b + 9$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

$1 = - 2 b + 9$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

$1 = - 2 b + 9$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

$1 = - 2 b + 9$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

Solve for $b$ in $1 = - 2 b + 9$ .

Rewrite the equation as $- 2 b + 9 = 1$ .

$- 2 b + 9 = 1$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

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

Subtract $9$ from both sides of the equation.

$- 2 b = 1 - 9$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

Subtract $9$ from $1$ .

$- 2 b = - 8$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

$- 2 b = - 8$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

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

Divide each term in $- 2 b = - 8$ by $- 2$ .

$\frac{- 2 b}{- 2} = \frac{- 8}{- 2}$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 b}{- 2} = \frac{- 8}{- 2}$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

Divide $b$ by $1$ .

$b = \frac{- 8}{- 2}$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

$b = \frac{- 8}{- 2}$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

$b = \frac{- 8}{- 2}$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

Simplify the right side.

Divide $- 8$ by $- 2$ .

$b = 4$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

$b = 4$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

$b = 4$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

$b = 4$

$27 = - 2 b + 27$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

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

Replace all occurrences of $b$ in $27 = - 2 b + 27$ with $4$ .

$27 = - 2 \cdot 4 + 27$

$b = 4$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

Simplify the right side.

Simplify $- 2 \cdot 4 + 27$ .

Multiply $- 2$ by $4$ .

$27 = - 8 + 27$

$b = 4$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

Add $- 8$ and $27$ .

$27 = 19$

$b = 4$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

$27 = 19$

$b = 4$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

$27 = 19$

$b = 4$

$81 = - 6 b + 57$

$a = - b + 6$

$c = 3$

Replace all occurrences of $b$ in $81 = - 6 b + 57$ with $4$ .

$81 = - 6 \cdot 4 + 57$

$27 = 19$

$b = 4$

$a = - b + 6$

$c = 3$

Simplify the right side.

Simplify $- 6 \cdot 4 + 57$ .

Multiply $- 6$ by $4$ .

$81 = - 24 + 57$

$27 = 19$

$b = 4$

$a = - b + 6$

$c = 3$

Add $- 24$ and $57$ .

$81 = 33$

$27 = 19$

$b = 4$

$a = - b + 6$

$c = 3$

$81 = 33$

$27 = 19$

$b = 4$

$a = - b + 6$

$c = 3$

$81 = 33$

$27 = 19$

$b = 4$

$a = - b + 6$

$c = 3$

Replace all occurrences of $b$ in $a = - b + 6$ with $4$ .

$a = - (4) + 6$

$81 = 33$

$27 = 19$

$b = 4$

$c = 3$

Simplify the right side.

Simplify $- (4) + 6$ .

Multiply $- 1$ by $4$ .

$a = - 4 + 6$

$81 = 33$

$27 = 19$

$b = 4$

$c = 3$

Add $- 4$ and $6$ .

$a = 2$

$81 = 33$

$27 = 19$

$b = 4$

$c = 3$

$a = 2$

$81 = 33$

$27 = 19$

$b = 4$

$c = 3$

$a = 2$

$81 = 33$

$27 = 19$

$b = 4$

$c = 3$

$a = 2$

$81 = 33$

$27 = 19$

$b = 4$

$c = 3$

Since $81 = 33$ is not true, there is no solution.

No solution

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

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 0$ , $b = 0$ , $c = 0$ , and $x = - 1$ .

Simplify each term.

Raise $- 1$ to the power of $2$ .

$y = 0 \cdot 1 + (0) \cdot (- 1) + 0$

Multiply $0$ by $1$ .

$y = 0 + (0) \cdot (- 1) + 0$

Multiply $0$ by $- 1$ .

$y = 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = - 1$ . This check does not pass, since $y = 0$ and $y = 1$ . The function rule can\'t be quadratic.

$0 \neq 1$

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

The function is not quadratic

Check if the function rule is cubic.

To find if the table follows a function rule, check whether the function rule could follow the form $y = a x^{3} + b x^{2} + c x + d$ .

$y = a x^{3} + b x^{2} + c x + d$

Build a set of $4$ equations from the table such that $y = a x^{3} + b x^{2} + c x + d$ .

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

Solve for $b$ in $1 = a {(- 1)}^{3} + b {(- 1)}^{2} + c (- 1) + d$ .

Rewrite the equation as $a {(- 1)}^{3} + b {(- 1)}^{2} + c (- 1) + d = 1$ .

$a {(- 1)}^{3} + b {(- 1)}^{2} + c (- 1) + d = 1$

$3 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Simplify each term.

Raise $- 1$ to the power of $3$ .

$a \cdot - 1 + b {(- 1)}^{2} + c (- 1) + d = 1$

$3 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

$- 1 \cdot a + b {(- 1)}^{2} + c (- 1) + d = 1$

$3 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

$- a + b {(- 1)}^{2} + c (- 1) + d = 1$

$3 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Raise $- 1$ to the power of $2$ .

$- a + b \cdot 1 + c (- 1) + d = 1$

$3 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Multiply $b$ by $1$ .

$- a + b + c (- 1) + d = 1$

$3 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

$- a + b - 1 c + d = 1$

$3 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Rewrite $- 1 c$ as $- c$ .

$- a + b - c + d = 1$

$3 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$- a + b - c + d = 1$

$3 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

Add $a$ to both sides of the equation.

$b - c + d = 1 + a$

$3 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Add $c$ to both sides of the equation.

$b + d = 1 + a + c$

$3 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Subtract $d$ from both sides of the equation.

$b = 1 + a + c - d$

$3 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$b = 1 + a + c - d$

$3 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$b = 1 + a + c - d$

$3 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Replace all occurrences of $b$ with $1 + a + c - d$ in each equation.

Replace all occurrences of $b$ in $3 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$ with $1 + a + c - d$ .

$3 = a \cdot 0^{3} + (1 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Simplify the right side.

Simplify $a \cdot 0^{3} + (1 + a + c - d) {(- 1)}^{2} + c (- 1) + d$ .

Simplify each term.

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

$3 = a \cdot 0 + (1 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Multiply $a$ by $0$ .

$3 = 0 + (1 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Raise $- 1$ to the power of $2$ .

$3 = 0 + (1 + a + c - d) \cdot 1 + c (- 1) + d$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Multiply $1 + a + c - d$ by $1$ .

$3 = 0 + 1 + a + c - d + c (- 1) + d$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

$3 = 0 + 1 + a + c - d - 1 c + d$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Rewrite $- 1 c$ as $- c$ .

$3 = 0 + 1 + a + c - d - c + d$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$3 = 0 + 1 + a + c - d - c + d$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Combine the opposite terms in $0 + 1 + a + c - d - c + d$ .

Add $0$ and $1$ .

$3 = 1 + a + c - d - c + d$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Subtract $c$ from $c$ .

$3 = 1 + a - d + 0 + d$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Add $1 + a - d$ and $0$ .

$3 = 1 + a - d + d$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Add $- d$ and $d$ .

$3 = 1 + a + 0$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Add $1 + a$ and $0$ .

$3 = 1 + a$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$3 = 1 + a$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$3 = 1 + a$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$3 = 1 + a$

$b = 1 + a + c - d$

$9 = a + b {(- 1)}^{2} + c (- 1) + d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Replace all occurrences of $b$ in $9 = a + b {(- 1)}^{2} + c (- 1) + d$ with $1 + a + c - d$ .

$9 = a + (1 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Simplify the right side.

Simplify $a + (1 + a + c - d) {(- 1)}^{2} + c (- 1) + d$ .

Simplify each term.

Raise $- 1$ to the power of $2$ .

$9 = a + (1 + a + c - d) \cdot 1 + c (- 1) + d$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Multiply $1 + a + c - d$ by $1$ .

$9 = a + 1 + a + c - d + c (- 1) + d$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

$9 = a + 1 + a + c - d - 1 c + d$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Rewrite $- 1 c$ as $- c$ .

$9 = a + 1 + a + c - d - c + d$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + 1 + a + c - d - c + d$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Simplify by adding terms.

Combine the opposite terms in $a + 1 + a + c - d - c + d$ .

Subtract $c$ from $c$ .

$9 = a + 1 + a - d + 0 + d$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Add $a + 1 + a - d$ and $0$ .

$9 = a + 1 + a - d + d$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Add $- d$ and $d$ .

$9 = a + 1 + a + 0$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Add $a + 1 + a$ and $0$ .

$9 = a + 1 + a$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = a + 1 + a$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Add $a$ and $a$ .

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Replace all occurrences of $b$ in $27 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$ with $1 + a + c - d$ .

$27 = a \cdot 2^{3} + (1 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Simplify the right side.

Simplify $a \cdot 2^{3} + (1 + a + c - d) {(- 1)}^{2} + c (- 1) + d$ .

Simplify each term.

Raise $2$ to the power of $3$ .

$27 = a \cdot 8 + (1 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Move $8$ to the left of $a$ .

$27 = 8 \cdot a + (1 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Raise $- 1$ to the power of $2$ .

$27 = 8 a + (1 + a + c - d) \cdot 1 + c (- 1) + d$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Multiply $1 + a + c - d$ by $1$ .

$27 = 8 a + 1 + a + c - d + c (- 1) + d$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

$27 = 8 a + 1 + a + c - d - 1 c + d$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Rewrite $- 1 c$ as $- c$ .

$27 = 8 a + 1 + a + c - d - c + d$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$27 = 8 a + 1 + a + c - d - c + d$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Simplify by adding terms.

Combine the opposite terms in $8 a + 1 + a + c - d - c + d$ .

Subtract $c$ from $c$ .

$27 = 8 a + 1 + a - d + 0 + d$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Add $8 a + 1 + a - d$ and $0$ .

$27 = 8 a + 1 + a - d + d$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Add $- d$ and $d$ .

$27 = 8 a + 1 + a + 0$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Add $8 a + 1 + a$ and $0$ .

$27 = 8 a + 1 + a$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$27 = 8 a + 1 + a$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Add $8 a$ and $a$ .

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$

Replace all occurrences of $b$ in $81 = a \cdot 3^{3} + b {(- 1)}^{2} + c (- 1) + d$ with $1 + a + c - d$ .

$81 = a \cdot 3^{3} + (1 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Simplify the right side.

Simplify $a \cdot 3^{3} + (1 + a + c - d) {(- 1)}^{2} + c (- 1) + d$ .

Simplify each term.

Raise $3$ to the power of $3$ .

$81 = a \cdot 27 + (1 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Move $27$ to the left of $a$ .

$81 = 27 \cdot a + (1 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Raise $- 1$ to the power of $2$ .

$81 = 27 a + (1 + a + c - d) \cdot 1 + c (- 1) + d$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Multiply $1 + a + c - d$ by $1$ .

$81 = 27 a + 1 + a + c - d + c (- 1) + d$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

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

$81 = 27 a + 1 + a + c - d - 1 c + d$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Rewrite $- 1 c$ as $- c$ .

$81 = 27 a + 1 + a + c - d - c + d$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = 27 a + 1 + a + c - d - c + d$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Simplify by adding terms.

Combine the opposite terms in $27 a + 1 + a + c - d - c + d$ .

Subtract $c$ from $c$ .

$81 = 27 a + 1 + a - d + 0 + d$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Add $27 a + 1 + a - d$ and $0$ .

$81 = 27 a + 1 + a - d + d$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Add $- d$ and $d$ .

$81 = 27 a + 1 + a + 0$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Add $27 a + 1 + a$ and $0$ .

$81 = 27 a + 1 + a$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = 27 a + 1 + a$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Add $27 a$ and $a$ .

$81 = 28 a + 1$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = 28 a + 1$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = 28 a + 1$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = 28 a + 1$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$81 = 28 a + 1$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Solve for $a$ in $81 = 28 a + 1$ .

Rewrite the equation as $28 a + 1 = 81$ .

$28 a + 1 = 81$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

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

Subtract $1$ from both sides of the equation.

$28 a = 81 - 1$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Subtract $1$ from $81$ .

$28 a = 80$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$28 a = 80$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Divide each term in $28 a = 80$ by $28$ and simplify.

Divide each term in $28 a = 80$ by $28$ .

$\frac{28 a}{28} = \frac{80}{28}$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Simplify the left side.

Cancel the common factor of $28$ .

Cancel the common factor.

$\frac{28 a}{28} = \frac{80}{28}$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Divide $a$ by $1$ .

$a = \frac{80}{28}$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$a = \frac{80}{28}$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$a = \frac{80}{28}$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Simplify the right side.

Cancel the common factor of $80$ and $28$ .

Factor $4$ out of $80$ .

$a = \frac{4 (20)}{28}$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Cancel the common factors.

Factor $4$ out of $28$ .

$a = \frac{4 \cdot 20}{4 \cdot 7}$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Cancel the common factor.

$a = \frac{4 \cdot 20}{4 \cdot 7}$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Rewrite the expression.

$a = \frac{20}{7}$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$a = \frac{20}{7}$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$a = \frac{20}{7}$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$a = \frac{20}{7}$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$a = \frac{20}{7}$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$a = \frac{20}{7}$

$27 = 9 a + 1$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Replace all occurrences of $a$ with $\frac{20}{7}$ in each equation.

Replace all occurrences of $a$ in $27 = 9 a + 1$ with $\frac{20}{7}$ .

$27 = 9 (\frac{20}{7}) + 1$

$a = \frac{20}{7}$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Simplify the right side.

Simplify $9 (\frac{20}{7}) + 1$ .

Multiply $9 (\frac{20}{7})$ .

Combine $9$ and $\frac{20}{7}$ .

$27 = \frac{9 \cdot 20}{7} + 1$

$a = \frac{20}{7}$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Multiply $9$ by $20$ .

$27 = \frac{180}{7} + 1$

$a = \frac{20}{7}$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = \frac{180}{7} + 1$

$a = \frac{20}{7}$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Write $1$ as a fraction with a common denominator.

$27 = \frac{180}{7} + \frac{7}{7}$

$a = \frac{20}{7}$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Combine the numerators over the common denominator.

$27 = \frac{180 + 7}{7}$

$a = \frac{20}{7}$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Add $180$ and $7$ .

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$9 = 2 a + 1$

$3 = 1 + a$

$b = 1 + a + c - d$

Replace all occurrences of $a$ in $9 = 2 a + 1$ with $\frac{20}{7}$ .

$9 = 2 (\frac{20}{7}) + 1$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$3 = 1 + a$

$b = 1 + a + c - d$

Simplify the right side.

Simplify $2 (\frac{20}{7}) + 1$ .

Multiply $2 (\frac{20}{7})$ .

Combine $2$ and $\frac{20}{7}$ .

$9 = \frac{2 \cdot 20}{7} + 1$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$3 = 1 + a$

$b = 1 + a + c - d$

Multiply $2$ by $20$ .

$9 = \frac{40}{7} + 1$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$3 = 1 + a$

$b = 1 + a + c - d$

$9 = \frac{40}{7} + 1$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$3 = 1 + a$

$b = 1 + a + c - d$

Write $1$ as a fraction with a common denominator.

$9 = \frac{40}{7} + \frac{7}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$3 = 1 + a$

$b = 1 + a + c - d$

Combine the numerators over the common denominator.

$9 = \frac{40 + 7}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$3 = 1 + a$

$b = 1 + a + c - d$

Add $40$ and $7$ .

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$3 = 1 + a$

$b = 1 + a + c - d$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$3 = 1 + a$

$b = 1 + a + c - d$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$3 = 1 + a$

$b = 1 + a + c - d$

Replace all occurrences of $a$ in $3 = 1 + a$ with $\frac{20}{7}$ .

$3 = 1 + \frac{20}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$b = 1 + a + c - d$

Simplify $3 = 1 + \frac{20}{7}$ .

Simplify the left side.

Remove parentheses.

$3 = 1 + \frac{20}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$b = 1 + a + c - d$

$3 = 1 + \frac{20}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$b = 1 + a + c - d$

Simplify the right side.

Simplify $1 + \frac{20}{7}$ .

Write $1$ as a fraction with a common denominator.

$3 = \frac{7}{7} + \frac{20}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$b = 1 + a + c - d$

Combine the numerators over the common denominator.

$3 = \frac{7 + 20}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$b = 1 + a + c - d$

Add $7$ and $20$ .

$3 = \frac{27}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$b = 1 + a + c - d$

$3 = \frac{27}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$b = 1 + a + c - d$

$3 = \frac{27}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$b = 1 + a + c - d$

$3 = \frac{27}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$b = 1 + a + c - d$

Replace all occurrences of $a$ in $b = 1 + a + c - d$ with $\frac{20}{7}$ .

$b = 1 + \frac{20}{7} + c - d$

$3 = \frac{27}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

Simplify $b = 1 + \frac{20}{7} + c - d$ .

Simplify the left side.

Remove parentheses.

$b = 1 + \frac{20}{7} + c - d$

$3 = \frac{27}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$b = 1 + \frac{20}{7} + c - d$

$3 = \frac{27}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

Simplify the right side.

Simplify $1 + \frac{20}{7} + c - d$ .

Write $1$ as a fraction with a common denominator.

$b = \frac{7}{7} + \frac{20}{7} + c - d$

$3 = \frac{27}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

Combine the numerators over the common denominator.

$b = \frac{7 + 20}{7} + c - d$

$3 = \frac{27}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

Add $7$ and $20$ .

$b = \frac{27}{7} + c - d$

$3 = \frac{27}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$b = \frac{27}{7} + c - d$

$3 = \frac{27}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$b = \frac{27}{7} + c - d$

$3 = \frac{27}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$b = \frac{27}{7} + c - d$

$3 = \frac{27}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

$b = \frac{27}{7} + c - d$

$3 = \frac{27}{7}$

$9 = \frac{47}{7}$

$27 = \frac{187}{7}$

$a = \frac{20}{7}$

Since $3 = \frac{27}{7}$ is not true, there is no solution.

No solution

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

Calculate the value of $y$ such that $y = a x^{3} + b$ when $a = 0$ , $b = 0$ , $c = 0$ , $d = 0$ , and $x = - 1$ .

Simplify each term.

Raise $- 1$ to the power of $3$ .

$y = 0 \cdot - 1 + (0) \cdot (- 1^{2}) + (0) \cdot ((- 1) \cdot 0)$

Multiply $0$ by $- 1$ .

$y = 0 + (0) \cdot (- 1^{2}) + (0) \cdot ((- 1) \cdot 0)$

One to any power is one.

$y = 0 + 0 \cdot (- 1 \cdot 1) + (0) \cdot ((- 1) \cdot 0)$

Multiply $- 1$ by $1$ .

$y = 0 + 0 \cdot - 1 + (0) \cdot ((- 1) \cdot 0)$

Multiply $0$ by $- 1$ .

$y = 0 + 0 + (0) \cdot ((- 1) \cdot 0)$

Multiply $- 1$ by $0$ .

$y = 0 + 0 + 0 \cdot 0$

Multiply $0$ by $0$ .

$y = 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

If the table has a cubic function rule, $y = y$ for the corresponding $x$ value, $x = - 1$ . This check does not pass, since $y = 0$ and $y = 1$ . The function rule can\'t be cubic.

$0 \neq 1$

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

The function is not cubic

There are no values for $a$ , $b$ , $c$ , and $d$ in the equations $y = a x + b$ , $y = a x^{2} + b x + c$ , and $y = a x^{3} + b x^{2} + c x + d$ that work for every pair of $x$ and $y$ .

The table does not have a function rule that is linear, quadratic, or cubic.

Find the Function Rule table[[x,y],[-1,1],[0,3],[1,9],[2,27],[3,81]]

Related Questions

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