Pre-Algebra Examples

$\begin{array}{c} x & y \\ - 1 & - 9 \\ 0 & - 5 \\ 1 & - 1 \\ 2 & - 1 \end{array}$

Step 1

Check if the function rule is linear.

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Step 1.1

To find if the table follows a function rule, check to see if the values follow the linear form $y = a x + b$ .

$y = a x + b$

Step 1.2

Build a set of equations from the table such that $y = a x + b$ .

$- 9 = a (- 1) + b - 5 = a (0) + b - 1 = a (1) + b - 1 = a (2) + b$

Step 1.3

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

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Step 1.3.1

Rewrite the equation as $b = - 5$ .

$b = - 5$

$- 9 = a (- 1) + b$

$- 1 = a + b$

$- 1 = a (2) + b$

Step 1.3.2

Replace all occurrences of $b$ with $- 5$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $- 9 = a (- 1) + b$ with $- 5$ .

$- 9 = a (- 1) - 5$

$b = - 5$

$- 1 = a + b$

$- 1 = a (2) + b$

Step 1.3.2.2

Simplify $- 9 = a (- 1) - 5$ .

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Step 1.3.2.2.1

Simplify the left side.

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Step 1.3.2.2.1.1

Remove parentheses.

$- 9 = a (- 1) - 5$

$b = - 5$

$- 1 = a + b$

$- 1 = a (2) + b$

$- 9 = a (- 1) - 5$

$b = - 5$

$- 1 = a + b$

$- 1 = a (2) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Simplify each term.

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Step 1.3.2.2.2.1.1

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

$- 9 = - 1 a - 5$

$b = - 5$

$- 1 = a + b$

$- 1 = a (2) + b$

Step 1.3.2.2.2.1.2

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

$- 9 = - a - 5$

$b = - 5$

$- 1 = a + b$

$- 1 = a (2) + b$

$- 9 = - a - 5$

$b = - 5$

$- 1 = a + b$

$- 1 = a (2) + b$

$- 9 = - a - 5$

$b = - 5$

$- 1 = a + b$

$- 1 = a (2) + b$

$- 9 = - a - 5$

$b = - 5$

$- 1 = a + b$

$- 1 = a (2) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $- 1 = a + b$ with $- 5$ .

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

$- 1 = a (2) + b$

Step 1.3.2.4

Simplify the left side.

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Step 1.3.2.4.1

Remove parentheses.

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

$- 1 = a (2) + b$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

$- 1 = a (2) + b$

Step 1.3.2.5

Replace all occurrences of $b$ in $- 1 = a (2) + b$ with $- 5$ .

$- 1 = a (2) - 5$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

Step 1.3.2.6

Simplify $- 1 = a (2) - 5$ .

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Step 1.3.2.6.1

Simplify the left side.

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Step 1.3.2.6.1.1

Remove parentheses.

$- 1 = a (2) - 5$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

$- 1 = a (2) - 5$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

Step 1.3.2.6.2

Simplify the right side.

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Step 1.3.2.6.2.1

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

$- 1 = 2 a - 5$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

$- 1 = 2 a - 5$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

$- 1 = 2 a - 5$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

$- 1 = 2 a - 5$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

Step 1.3.3

Solve for $a$ in $- 1 = 2 a - 5$ .

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Step 1.3.3.1

Rewrite the equation as $2 a - 5 = - 1$ .

$2 a - 5 = - 1$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

Step 1.3.3.2

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

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Step 1.3.3.2.1

Add $5$ to both sides of the equation.

$2 a = - 1 + 5$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

Step 1.3.3.2.2

Add $- 1$ and $5$ .

$2 a = 4$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

$2 a = 4$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

Step 1.3.3.3

Divide each term in $2 a = 4$ by $2$ and simplify.

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Step 1.3.3.3.1

Divide each term in $2 a = 4$ by $2$ .

$\frac{2 a}{2} = \frac{4}{2}$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

Step 1.3.3.3.2

Simplify the left side.

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Step 1.3.3.3.2.1

Cancel the common factor of $2$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{4}{2}$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{4}{2}$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

$a = \frac{4}{2}$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

$a = \frac{4}{2}$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide $4$ by $2$ .

$a = 2$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

$a = 2$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

$a = 2$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

$a = 2$

$- 1 = a - 5$

$- 9 = - a - 5$

$b = - 5$

Step 1.3.4

Replace all occurrences of $a$ with $2$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $a$ in $- 1 = a - 5$ with $2$ .

$- 1 = (2) - 5$

$a = 2$

$- 9 = - a - 5$

$b = - 5$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Subtract $5$ from $2$ .

$- 1 = - 3$

$a = 2$

$- 9 = - a - 5$

$b = - 5$

$- 1 = - 3$

$a = 2$

$- 9 = - a - 5$

$b = - 5$

Step 1.3.4.3

Replace all occurrences of $a$ in $- 9 = - a - 5$ with $2$ .

$- 9 = - (2) - 5$

$- 1 = - 3$

$a = 2$

$b = - 5$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $- (2) - 5$ .

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Step 1.3.4.4.1.1

Multiply $- 1$ by $2$ .

$- 9 = - 2 - 5$

$- 1 = - 3$

$a = 2$

$b = - 5$

Step 1.3.4.4.1.2

Subtract $5$ from $- 2$ .

$- 9 = - 7$

$- 1 = - 3$

$a = 2$

$b = - 5$

$- 9 = - 7$

$- 1 = - 3$

$a = 2$

$b = - 5$

$- 9 = - 7$

$- 1 = - 3$

$a = 2$

$b = - 5$

$- 9 = - 7$

$- 1 = - 3$

$a = 2$

$b = - 5$

Step 1.3.5

Since $- 9 = - 7$ is not true, there is no solution.

No solution

Step 1.4

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.

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Step 2.1

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

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

Step 2.2

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

Step 2.3

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

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Step 2.3.1

Solve for $c$ in $- 5 = a \cdot 0^{2} + c$ .

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Step 2.3.1.1

Rewrite the equation as $a \cdot 0^{2} + c = - 5$ .

$a \cdot 0^{2} + c = - 5$

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

$- 1 = a + b + c$

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

Step 2.3.1.2

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

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Step 2.3.1.2.1

Simplify each term.

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Step 2.3.1.2.1.1

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

$a \cdot 0 + c = - 5$

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

$- 1 = a + b + c$

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

Step 2.3.1.2.1.2

Multiply $a$ by $0$ .

$0 + c = - 5$

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

$- 1 = a + b + c$

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

$0 + c = - 5$

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

$- 1 = a + b + c$

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

Step 2.3.1.2.2

Add $0$ and $c$ .

$c = - 5$

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

$- 1 = a + b + c$

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

$c = - 5$

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

$- 1 = a + b + c$

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

$c = - 5$

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

$- 1 = a + b + c$

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

Step 2.3.2

Replace all occurrences of $c$ with $- 5$ in each equation.

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Step 2.3.2.1

Replace all occurrences of $c$ in $- 9 = a {(- 1)}^{2} + b (- 1) + c$ with $- 5$ .

$- 9 = a {(- 1)}^{2} + b (- 1) - 5$

$c = - 5$

$- 1 = a + b + c$

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

Step 2.3.2.2

Simplify $- 9 = a {(- 1)}^{2} + b (- 1) - 5$ .

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Step 2.3.2.2.1

Simplify the left side.

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Step 2.3.2.2.1.1

Remove parentheses.

$- 9 = a {(- 1)}^{2} + b (- 1) - 5$

$c = - 5$

$- 1 = a + b + c$

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

$- 9 = a {(- 1)}^{2} + b (- 1) - 5$

$c = - 5$

$- 1 = a + b + c$

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

Step 2.3.2.2.2

Simplify the right side.

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Step 2.3.2.2.2.1

Simplify each term.

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Step 2.3.2.2.2.1.1

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

$- 9 = a \cdot 1 + b (- 1) - 5$

$c = - 5$

$- 1 = a + b + c$

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

Step 2.3.2.2.2.1.2

Multiply $a$ by $1$ .

$- 9 = a + b (- 1) - 5$

$c = - 5$

$- 1 = a + b + c$

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

Step 2.3.2.2.2.1.3

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

$- 9 = a - 1 b - 5$

$c = - 5$

$- 1 = a + b + c$

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

Step 2.3.2.2.2.1.4

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

$- 9 = a - b - 5$

$c = - 5$

$- 1 = a + b + c$

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

$- 9 = a - b - 5$

$c = - 5$

$- 1 = a + b + c$

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

$- 9 = a - b - 5$

$c = - 5$

$- 1 = a + b + c$

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

$- 9 = a - b - 5$

$c = - 5$

$- 1 = a + b + c$

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

Step 2.3.2.3

Replace all occurrences of $c$ in $- 1 = a + b + c$ with $- 5$ .

$- 1 = a + b - 5$

$- 9 = a - b - 5$

$c = - 5$

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

Step 2.3.2.4

Simplify the left side.

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Step 2.3.2.4.1

Remove parentheses.

$- 1 = a + b - 5$

$- 9 = a - b - 5$

$c = - 5$

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

$- 1 = a + b - 5$

$- 9 = a - b - 5$

$c = - 5$

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

Step 2.3.2.5

Replace all occurrences of $c$ in $- 1 = a \cdot 2^{2} + b (2) + c$ with $- 5$ .

$- 1 = a \cdot 2^{2} + b (2) - 5$

$- 1 = a + b - 5$

$- 9 = a - b - 5$

$c = - 5$

Step 2.3.2.6

Simplify $- 1 = a \cdot 2^{2} + b (2) - 5$ .

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Step 2.3.2.6.1

Simplify the left side.

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Step 2.3.2.6.1.1

Remove parentheses.

$- 1 = a \cdot 2^{2} + b (2) - 5$

$- 1 = a + b - 5$

$- 9 = a - b - 5$

$c = - 5$

$- 1 = a \cdot 2^{2} + b (2) - 5$

$- 1 = a + b - 5$

$- 9 = a - b - 5$

$c = - 5$

Step 2.3.2.6.2

Simplify the right side.

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Step 2.3.2.6.2.1

Simplify each term.

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Step 2.3.2.6.2.1.1

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

$- 1 = a \cdot 4 + b (2) - 5$

$- 1 = a + b - 5$

$- 9 = a - b - 5$

$c = - 5$

Step 2.3.2.6.2.1.2

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

$- 1 = 4 \cdot a + b (2) - 5$

$- 1 = a + b - 5$

$- 9 = a - b - 5$

$c = - 5$

Step 2.3.2.6.2.1.3

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

$- 1 = 4 a + 2 b - 5$

$- 1 = a + b - 5$

$- 9 = a - b - 5$

$c = - 5$

$- 1 = 4 a + 2 b - 5$

$- 1 = a + b - 5$

$- 9 = a - b - 5$

$c = - 5$

$- 1 = 4 a + 2 b - 5$

$- 1 = a + b - 5$

$- 9 = a - b - 5$

$c = - 5$

$- 1 = 4 a + 2 b - 5$

$- 1 = a + b - 5$

$- 9 = a - b - 5$

$c = - 5$

$- 1 = 4 a + 2 b - 5$

$- 1 = a + b - 5$

$- 9 = a - b - 5$

$c = - 5$

Step 2.3.3

Solve for $a$ in $- 1 = a + b - 5$ .

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Step 2.3.3.1

Rewrite the equation as $a + b - 5 = - 1$ .

$a + b - 5 = - 1$

$- 1 = 4 a + 2 b - 5$

$- 9 = a - b - 5$

$c = - 5$

Step 2.3.3.2

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

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Step 2.3.3.2.1

Subtract $b$ from both sides of the equation.

$a - 5 = - 1 - b$

$- 1 = 4 a + 2 b - 5$

$- 9 = a - b - 5$

$c = - 5$

Step 2.3.3.2.2

Add $5$ to both sides of the equation.

$a = - 1 - b + 5$

$- 1 = 4 a + 2 b - 5$

$- 9 = a - b - 5$

$c = - 5$

Step 2.3.3.2.3

Add $- 1$ and $5$ .

$a = - b + 4$

$- 1 = 4 a + 2 b - 5$

$- 9 = a - b - 5$

$c = - 5$

$a = - b + 4$

$- 1 = 4 a + 2 b - 5$

$- 9 = a - b - 5$

$c = - 5$

$a = - b + 4$

$- 1 = 4 a + 2 b - 5$

$- 9 = a - b - 5$

$c = - 5$

Step 2.3.4

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

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Step 2.3.4.1

Replace all occurrences of $a$ in $- 1 = 4 a + 2 b - 5$ with $- b + 4$ .

$- 1 = 4 (- b + 4) + 2 b - 5$

$a = - b + 4$

$- 9 = a - b - 5$

$c = - 5$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $4 (- b + 4) + 2 b - 5$ .

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Step 2.3.4.2.1.1

Simplify each term.

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Step 2.3.4.2.1.1.1

Apply the distributive property.

$- 1 = 4 (- b) + 4 \cdot 4 + 2 b - 5$

$a = - b + 4$

$- 9 = a - b - 5$

$c = - 5$

Step 2.3.4.2.1.1.2

Multiply $- 1$ by $4$ .

$- 1 = - 4 b + 4 \cdot 4 + 2 b - 5$

$a = - b + 4$

$- 9 = a - b - 5$

$c = - 5$

Step 2.3.4.2.1.1.3

Multiply $4$ by $4$ .

$- 1 = - 4 b + 16 + 2 b - 5$

$a = - b + 4$

$- 9 = a - b - 5$

$c = - 5$

$- 1 = - 4 b + 16 + 2 b - 5$

$a = - b + 4$

$- 9 = a - b - 5$

$c = - 5$

Step 2.3.4.2.1.2

Simplify by adding terms.

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Step 2.3.4.2.1.2.1

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

$- 1 = - 2 b + 16 - 5$

$a = - b + 4$

$- 9 = a - b - 5$

$c = - 5$

Step 2.3.4.2.1.2.2

Subtract $5$ from $16$ .

$- 1 = - 2 b + 11$

$a = - b + 4$

$- 9 = a - b - 5$

$c = - 5$

$- 1 = - 2 b + 11$

$a = - b + 4$

$- 9 = a - b - 5$

$c = - 5$

$- 1 = - 2 b + 11$

$a = - b + 4$

$- 9 = a - b - 5$

$c = - 5$

$- 1 = - 2 b + 11$

$a = - b + 4$

$- 9 = a - b - 5$

$c = - 5$

Step 2.3.4.3

Replace all occurrences of $a$ in $- 9 = a - b - 5$ with $- b + 4$ .

$- 9 = (- b + 4) - b - 5$

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $(- b + 4) - b - 5$ .

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Step 2.3.4.4.1.1

Subtract $b$ from $- b$ .

$- 9 = - 2 b + 4 - 5$

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

Step 2.3.4.4.1.2

Subtract $5$ from $4$ .

$- 9 = - 2 b - 1$

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

$- 9 = - 2 b - 1$

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

$- 9 = - 2 b - 1$

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

$- 9 = - 2 b - 1$

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

Step 2.3.5

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

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Step 2.3.5.1

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

$- 2 b - 1 = - 9$

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

Step 2.3.5.2

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

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Step 2.3.5.2.1

Add $1$ to both sides of the equation.

$- 2 b = - 9 + 1$

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

Step 2.3.5.2.2

Add $- 9$ and $1$ .

$- 2 b = - 8$

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

$- 2 b = - 8$

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

Step 2.3.5.3

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

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Step 2.3.5.3.1

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

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

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

Step 2.3.5.3.2

Simplify the left side.

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Step 2.3.5.3.2.1

Cancel the common factor of $- 2$ .

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Step 2.3.5.3.2.1.1

Cancel the common factor.

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

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

Step 2.3.5.3.2.1.2

Divide $b$ by $1$ .

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

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

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

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

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

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

Step 2.3.5.3.3

Simplify the right side.

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Step 2.3.5.3.3.1

Divide $- 8$ by $- 2$ .

$b = 4$

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

$b = 4$

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

$b = 4$

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

$b = 4$

$- 1 = - 2 b + 11$

$a = - b + 4$

$c = - 5$

Step 2.3.6

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

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Step 2.3.6.1

Replace all occurrences of $b$ in $- 1 = - 2 b + 11$ with $4$ .

$- 1 = - 2 \cdot 4 + 11$

$b = 4$

$a = - b + 4$

$c = - 5$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $- 2 \cdot 4 + 11$ .

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Step 2.3.6.2.1.1

Multiply $- 2$ by $4$ .

$- 1 = - 8 + 11$

$b = 4$

$a = - b + 4$

$c = - 5$

Step 2.3.6.2.1.2

Add $- 8$ and $11$ .

$- 1 = 3$

$b = 4$

$a = - b + 4$

$c = - 5$

$- 1 = 3$

$b = 4$

$a = - b + 4$

$c = - 5$

$- 1 = 3$

$b = 4$

$a = - b + 4$

$c = - 5$

Step 2.3.6.3

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

$a = - (4) + 4$

$- 1 = 3$

$b = 4$

$c = - 5$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify $- (4) + 4$ .

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Step 2.3.6.4.1.1

Multiply $- 1$ by $4$ .

$a = - 4 + 4$

$- 1 = 3$

$b = 4$

$c = - 5$

Step 2.3.6.4.1.2

Add $- 4$ and $4$ .

$a = 0$

$- 1 = 3$

$b = 4$

$c = - 5$

$a = 0$

$- 1 = 3$

$b = 4$

$c = - 5$

$a = 0$

$- 1 = 3$

$b = 4$

$c = - 5$

$a = 0$

$- 1 = 3$

$b = 4$

$c = - 5$

Step 2.3.7

Since $- 1 = 3$ is not true, there is no solution.

No solution

Step 2.4

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

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Step 2.4.1

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

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

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

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

Step 2.4.1.1.2

Multiply $0$ by $1$ .

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

Step 2.4.1.1.3

Multiply $0$ by $- 1$ .

$y = 0 + 0 + 0$

Step 2.4.1.2

Simplify by adding numbers.

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Step 2.4.1.2.1

Add $0$ and $0$ .

$y = 0 + 0$

Step 2.4.1.2.2

Add $0$ and $0$ .

$y = 0$

Step 2.4.2

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 $y = - 9$ . The function rule can't be quadratic.

$0 \neq - 9$

Step 2.4.3

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

The function is not quadratic

Step 3

Check if the function rule is cubic.

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Step 3.1

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

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

Step 3.2

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

Step 3.3

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

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Step 3.3.1

Solve for $b$ in $- 9 = a {(- 1)}^{3} + b {(- 1)}^{2} + c (- 1) + d$ .

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Step 3.3.1.1

Rewrite the equation as $a {(- 1)}^{3} + b {(- 1)}^{2} + c (- 1) + d = - 9$ .

$a {(- 1)}^{3} + b {(- 1)}^{2} + c (- 1) + d = - 9$

$- 5 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

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

Step 3.3.1.2

Simplify each term.

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Step 3.3.1.2.1

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

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

$- 5 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

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

Step 3.3.1.2.2

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

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

$- 5 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

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

Step 3.3.1.2.3

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

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

$- 5 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

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

Step 3.3.1.2.4

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

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

$- 5 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

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

Step 3.3.1.2.5

Multiply $b$ by $1$ .

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

$- 5 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

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

Step 3.3.1.2.6

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

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

$- 5 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

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

Step 3.3.1.2.7

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

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

$- 5 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

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

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

$- 5 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

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

Step 3.3.1.3

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

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Step 3.3.1.3.1

Add $a$ to both sides of the equation.

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

$- 5 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

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

Step 3.3.1.3.2

Add $c$ to both sides of the equation.

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

$- 5 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

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

Step 3.3.1.3.3

Subtract $d$ from both sides of the equation.

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

$- 5 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

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

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

$- 5 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

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

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

$- 5 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$

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

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

Step 3.3.2

Replace all occurrences of $b$ with $- 9 + a + c - d$ in each equation.

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Step 3.3.2.1

Replace all occurrences of $b$ in $- 5 = a \cdot 0^{3} + b {(- 1)}^{2} + c (- 1) + d$ with $- 9 + a + c - d$ .

$- 5 = a \cdot 0^{3} + (- 9 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

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

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

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

Step 3.3.2.2

Simplify the right side.

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Step 3.3.2.2.1

Simplify $a \cdot 0^{3} + (- 9 + a + c - d) {(- 1)}^{2} + c (- 1) + d$ .

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Step 3.3.2.2.1.1

Simplify each term.

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Step 3.3.2.2.1.1.1

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

$- 5 = a \cdot 0 + (- 9 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

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

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

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

Step 3.3.2.2.1.1.2

Multiply $a$ by $0$ .

$- 5 = 0 + (- 9 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

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

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

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

Step 3.3.2.2.1.1.3

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

$- 5 = 0 + (- 9 + a + c - d) \cdot 1 + c (- 1) + d$

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

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

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

Step 3.3.2.2.1.1.4

Multiply $- 9 + a + c - d$ by $1$ .

$- 5 = 0 - 9 + a + c - d + c (- 1) + d$

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

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

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

Step 3.3.2.2.1.1.5

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

$- 5 = 0 - 9 + a + c - d - 1 c + d$

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

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

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

Step 3.3.2.2.1.1.6

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

$- 5 = 0 - 9 + a + c - d - c + d$

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

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

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

$- 5 = 0 - 9 + a + c - d - c + d$

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

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

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

Step 3.3.2.2.1.2

Combine the opposite terms in $0 - 9 + a + c - d - c + d$ .

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Step 3.3.2.2.1.2.1

Subtract $9$ from $0$ .

$- 5 = - 9 + a + c - d - c + d$

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

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

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

Step 3.3.2.2.1.2.2

Subtract $c$ from $c$ .

$- 5 = - 9 + a - d + 0 + d$

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

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

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

Step 3.3.2.2.1.2.3

Add $- 9 + a - d$ and $0$ .

$- 5 = - 9 + a - d + d$

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

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

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

Step 3.3.2.2.1.2.4

Add $- d$ and $d$ .

$- 5 = - 9 + a + 0$

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

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

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

Step 3.3.2.2.1.2.5

Add $- 9 + a$ and $0$ .

$- 5 = - 9 + a$

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

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

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

$- 5 = - 9 + a$

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

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

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

$- 5 = - 9 + a$

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

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

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

$- 5 = - 9 + a$

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

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

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

Step 3.3.2.3

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

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

$- 5 = - 9 + a$

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

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

Step 3.3.2.4

Simplify the right side.

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Step 3.3.2.4.1

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

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Step 3.3.2.4.1.1

Simplify each term.

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Step 3.3.2.4.1.1.1

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

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

$- 5 = - 9 + a$

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

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

Step 3.3.2.4.1.1.2

Multiply $- 9 + a + c - d$ by $1$ .

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

$- 5 = - 9 + a$

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

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

Step 3.3.2.4.1.1.3

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

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

$- 5 = - 9 + a$

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

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

Step 3.3.2.4.1.1.4

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

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

$- 5 = - 9 + a$

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

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

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

$- 5 = - 9 + a$

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

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

Step 3.3.2.4.1.2

Simplify by adding terms.

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Step 3.3.2.4.1.2.1

Combine the opposite terms in $a - 9 + a + c - d - c + d$ .

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Step 3.3.2.4.1.2.1.1

Subtract $c$ from $c$ .

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

$- 5 = - 9 + a$

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

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

Step 3.3.2.4.1.2.1.2

Add $a - 9 + a - d$ and $0$ .

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

$- 5 = - 9 + a$

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

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

Step 3.3.2.4.1.2.1.3

Add $- d$ and $d$ .

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

$- 5 = - 9 + a$

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

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

Step 3.3.2.4.1.2.1.4

Add $a - 9 + a$ and $0$ .

$- 1 = a - 9 + a$

$- 5 = - 9 + a$

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

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

$- 1 = a - 9 + a$

$- 5 = - 9 + a$

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

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

Step 3.3.2.4.1.2.2

Add $a$ and $a$ .

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

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

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

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

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

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

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

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

Step 3.3.2.5

Replace all occurrences of $b$ in $- 1 = a \cdot 2^{3} + b {(- 1)}^{2} + c (- 1) + d$ with $- 9 + a + c - d$ .

$- 1 = a \cdot 2^{3} + (- 9 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.2.6

Simplify the right side.

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Step 3.3.2.6.1

Simplify $a \cdot 2^{3} + (- 9 + a + c - d) {(- 1)}^{2} + c (- 1) + d$ .

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Step 3.3.2.6.1.1

Simplify each term.

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Step 3.3.2.6.1.1.1

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

$- 1 = a \cdot 8 + (- 9 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.2.6.1.1.2

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

$- 1 = 8 \cdot a + (- 9 + a + c - d) {(- 1)}^{2} + c (- 1) + d$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.2.6.1.1.3

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

$- 1 = 8 a + (- 9 + a + c - d) \cdot 1 + c (- 1) + d$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.2.6.1.1.4

Multiply $- 9 + a + c - d$ by $1$ .

$- 1 = 8 a - 9 + a + c - d + c (- 1) + d$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.2.6.1.1.5

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

$- 1 = 8 a - 9 + a + c - d - 1 c + d$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.2.6.1.1.6

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

$- 1 = 8 a - 9 + a + c - d - c + d$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

$- 1 = 8 a - 9 + a + c - d - c + d$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.2.6.1.2

Simplify by adding terms.

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Step 3.3.2.6.1.2.1

Combine the opposite terms in $8 a - 9 + a + c - d - c + d$ .

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Step 3.3.2.6.1.2.1.1

Subtract $c$ from $c$ .

$- 1 = 8 a - 9 + a - d + 0 + d$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.2.6.1.2.1.2

Add $8 a - 9 + a - d$ and $0$ .

$- 1 = 8 a - 9 + a - d + d$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.2.6.1.2.1.3

Add $- d$ and $d$ .

$- 1 = 8 a - 9 + a + 0$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.2.6.1.2.1.4

Add $8 a - 9 + a$ and $0$ .

$- 1 = 8 a - 9 + a$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

$- 1 = 8 a - 9 + a$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.2.6.1.2.2

Add $8 a$ and $a$ .

$- 1 = 9 a - 9$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

$- 1 = 9 a - 9$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

$- 1 = 9 a - 9$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

$- 1 = 9 a - 9$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

$- 1 = 9 a - 9$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.3

Solve for $a$ in $- 1 = 9 a - 9$ .

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Step 3.3.3.1

Rewrite the equation as $9 a - 9 = - 1$ .

$9 a - 9 = - 1$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.3.2

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

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Step 3.3.3.2.1

Add $9$ to both sides of the equation.

$9 a = - 1 + 9$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.3.2.2

Add $- 1$ and $9$ .

$9 a = 8$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

$9 a = 8$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.3.3

Divide each term in $9 a = 8$ by $9$ and simplify.

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Step 3.3.3.3.1

Divide each term in $9 a = 8$ by $9$ .

$\frac{9 a}{9} = \frac{8}{9}$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.3.3.2

Simplify the left side.

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Step 3.3.3.3.2.1

Cancel the common factor of $9$ .

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Step 3.3.3.3.2.1.1

Cancel the common factor.

$\frac{9 a}{9} = \frac{8}{9}$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{8}{9}$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

$a = \frac{8}{9}$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

$a = \frac{8}{9}$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

$a = \frac{8}{9}$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

$a = \frac{8}{9}$

$- 1 = 2 a - 9$

$- 5 = - 9 + a$

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

Step 3.3.4

Replace all occurrences of $a$ with $\frac{8}{9}$ in each equation.

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Step 3.3.4.1

Replace all occurrences of $a$ in $- 1 = 2 a - 9$ with $\frac{8}{9}$ .

$- 1 = 2 (\frac{8}{9}) - 9$

$a = \frac{8}{9}$

$- 5 = - 9 + a$

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

Step 3.3.4.2

Simplify the right side.

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Step 3.3.4.2.1

Simplify $2 (\frac{8}{9}) - 9$ .

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Step 3.3.4.2.1.1

Multiply $2 (\frac{8}{9})$ .

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Step 3.3.4.2.1.1.1

Combine $2$ and $\frac{8}{9}$ .

$- 1 = \frac{2 \cdot 8}{9} - 9$

$a = \frac{8}{9}$

$- 5 = - 9 + a$

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

Step 3.3.4.2.1.1.2

Multiply $2$ by $8$ .

$- 1 = \frac{16}{9} - 9$

$a = \frac{8}{9}$

$- 5 = - 9 + a$

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

$- 1 = \frac{16}{9} - 9$

$a = \frac{8}{9}$

$- 5 = - 9 + a$

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

Step 3.3.4.2.1.2

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- 1 = \frac{16}{9} - 9 \cdot \frac{9}{9}$

$a = \frac{8}{9}$

$- 5 = - 9 + a$

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

Step 3.3.4.2.1.3

Combine $- 9$ and $\frac{9}{9}$ .

$- 1 = \frac{16}{9} + \frac{- 9 \cdot 9}{9}$

$a = \frac{8}{9}$

$- 5 = - 9 + a$

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

Step 3.3.4.2.1.4

Combine the numerators over the common denominator.

$- 1 = \frac{16 - 9 \cdot 9}{9}$

$a = \frac{8}{9}$

$- 5 = - 9 + a$

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

Step 3.3.4.2.1.5

Simplify the numerator.

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Step 3.3.4.2.1.5.1

Multiply $- 9$ by $9$ .

$- 1 = \frac{16 - 81}{9}$

$a = \frac{8}{9}$

$- 5 = - 9 + a$

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

Step 3.3.4.2.1.5.2

Subtract $81$ from $16$ .

$- 1 = \frac{- 65}{9}$

$a = \frac{8}{9}$

$- 5 = - 9 + a$

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

$- 1 = \frac{- 65}{9}$

$a = \frac{8}{9}$

$- 5 = - 9 + a$

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

Step 3.3.4.2.1.6

Move the negative in front of the fraction.

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

$- 5 = - 9 + a$

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

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

$- 5 = - 9 + a$

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

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

$- 5 = - 9 + a$

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

Step 3.3.4.3

Replace all occurrences of $a$ in $- 5 = - 9 + a$ with $\frac{8}{9}$ .

$- 5 = - 9 + \frac{8}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

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

Step 3.3.4.4

Simplify $- 5 = - 9 + \frac{8}{9}$ .

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Step 3.3.4.4.1

Simplify the left side.

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Step 3.3.4.4.1.1

Remove parentheses.

$- 5 = - 9 + \frac{8}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

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

$- 5 = - 9 + \frac{8}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

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

Step 3.3.4.4.2

Simplify the right side.

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Step 3.3.4.4.2.1

Simplify $- 9 + \frac{8}{9}$ .

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Step 3.3.4.4.2.1.1

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- 5 = - 9 \cdot \frac{9}{9} + \frac{8}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

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

Step 3.3.4.4.2.1.2

Combine $- 9$ and $\frac{9}{9}$ .

$- 5 = \frac{- 9 \cdot 9}{9} + \frac{8}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

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

Step 3.3.4.4.2.1.3

Combine the numerators over the common denominator.

$- 5 = \frac{- 9 \cdot 9 + 8}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

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

Step 3.3.4.4.2.1.4

Simplify the numerator.

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Step 3.3.4.4.2.1.4.1

Multiply $- 9$ by $9$ .

$- 5 = \frac{- 81 + 8}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

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

Step 3.3.4.4.2.1.4.2

Add $- 81$ and $8$ .

$- 5 = \frac{- 73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

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

$- 5 = \frac{- 73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

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

Step 3.3.4.4.2.1.5

Move the negative in front of the fraction.

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

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

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

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

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

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

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

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

Step 3.3.4.5

Replace all occurrences of $a$ in $b = - 9 + a + c - d$ with $\frac{8}{9}$ .

$b = - 9 + \frac{8}{9} + c - d$

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

Step 3.3.4.6

Simplify $b = - 9 + \frac{8}{9} + c - d$ .

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Step 3.3.4.6.1

Simplify the left side.

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Step 3.3.4.6.1.1

Remove parentheses.

$b = - 9 + \frac{8}{9} + c - d$

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

$b = - 9 + \frac{8}{9} + c - d$

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

Step 3.3.4.6.2

Simplify the right side.

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Step 3.3.4.6.2.1

Simplify $- 9 + \frac{8}{9} + c - d$ .

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Step 3.3.4.6.2.1.1

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$b = - 9 \cdot \frac{9}{9} + \frac{8}{9} + c - d$

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

Step 3.3.4.6.2.1.2

Combine $- 9$ and $\frac{9}{9}$ .

$b = \frac{- 9 \cdot 9}{9} + \frac{8}{9} + c - d$

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

Step 3.3.4.6.2.1.3

Combine the numerators over the common denominator.

$b = \frac{- 9 \cdot 9 + 8}{9} + c - d$

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

Step 3.3.4.6.2.1.4

Simplify the numerator.

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Step 3.3.4.6.2.1.4.1

Multiply $- 9$ by $9$ .

$b = \frac{- 81 + 8}{9} + c - d$

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

Step 3.3.4.6.2.1.4.2

Add $- 81$ and $8$ .

$b = \frac{- 73}{9} + c - d$

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

$b = \frac{- 73}{9} + c - d$

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

Step 3.3.4.6.2.1.5

Move the negative in front of the fraction.

$b = - \frac{73}{9} + c - d$

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

$b = - \frac{73}{9} + c - d$

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

$b = - \frac{73}{9} + c - d$

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

$b = - \frac{73}{9} + c - d$

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

$b = - \frac{73}{9} + c - d$

$- 5 = - \frac{73}{9}$

$- 1 = - \frac{65}{9}$

$a = \frac{8}{9}$

Step 3.3.5

Since $- 5 = - \frac{73}{9}$ is not true, there is no solution.

No solution

Step 3.4

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

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Step 3.4.1

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

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Step 3.4.1.1

Simplify each term.

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Step 3.4.1.1.1

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

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

Step 3.4.1.1.2

Multiply $0$ by $- 1$ .

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

Step 3.4.1.1.3

One to any power is one.

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

Step 3.4.1.1.4

Multiply $0 (- 1 \cdot 1)$ .

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Step 3.4.1.1.4.1

Multiply $- 1$ by $1$ .

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

Step 3.4.1.1.4.2

Multiply $0$ by $- 1$ .

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

Step 3.4.1.1.5

Multiply $(0) ((- 1) \cdot 0)$ .

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Step 3.4.1.1.5.1

Multiply $- 1$ by $0$ .

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

Step 3.4.1.1.5.2

Multiply $0$ by $0$ .

$y = 0 + 0 + 0$

Step 3.4.1.2

Simplify by adding numbers.

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Step 3.4.1.2.1

Add $0$ and $0$ .

$y = 0 + 0$

Step 3.4.1.2.2

Add $0$ and $0$ .

$y = 0$

Step 3.4.2

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 $y = - 9$ . The function rule can't be cubic.

$0 \neq - 9$

Step 3.4.3

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

The function is not cubic

Step 4

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.