Pre-Algebra Examples

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

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 $y = a x + b$ .

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

Step 1.3

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

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Step 1.3.1

Rewrite the equation as $b = 3$ .

$b = 3$

$2 = a + b$

$4 = a (2) + b$

$1 = a (3) + b$

Step 1.3.2

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

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Step 1.3.2.1

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

$2 = a + 3$

$b = 3$

$4 = a (2) + b$

$1 = a (3) + b$

Step 1.3.2.2

Simplify the left side.

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Step 1.3.2.2.1

Remove parentheses.

$2 = a + 3$

$b = 3$

$4 = a (2) + b$

$1 = a (3) + b$

$2 = a + 3$

$b = 3$

$4 = a (2) + b$

$1 = a (3) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $4 = a (2) + b$ with $3$ .

$4 = a (2) + 3$

$2 = a + 3$

$b = 3$

$1 = a (3) + b$

Step 1.3.2.4

Simplify $4 = a (2) + 3$ .

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Step 1.3.2.4.1

Simplify the left side.

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Step 1.3.2.4.1.1

Remove parentheses.

$4 = a (2) + 3$

$2 = a + 3$

$b = 3$

$1 = a (3) + b$

$4 = a (2) + 3$

$2 = a + 3$

$b = 3$

$1 = a (3) + b$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

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

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

$1 = a (3) + b$

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

$1 = a (3) + b$

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

$1 = a (3) + b$

Step 1.3.2.5

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

$1 = a (3) + 3$

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

Step 1.3.2.6

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

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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.

$1 = a (3) + 3$

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

$1 = a (3) + 3$

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

Step 1.3.2.6.2

Simplify the right side.

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Step 1.3.2.6.2.1

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

$1 = 3 a + 3$

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

$1 = 3 a + 3$

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

$1 = 3 a + 3$

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

$1 = 3 a + 3$

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

Step 1.3.3

Solve for $a$ in $1 = 3 a + 3$ .

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Step 1.3.3.1

Rewrite the equation as $3 a + 3 = 1$ .

$3 a + 3 = 1$

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

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

Subtract $3$ from both sides of the equation.

$3 a = 1 - 3$

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

Step 1.3.3.2.2

Subtract $3$ from $1$ .

$3 a = - 2$

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

$3 a = - 2$

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

Step 1.3.3.3

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

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Step 1.3.3.3.1

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

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

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

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 $3$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

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

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

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

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

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

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

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

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Move the negative in front of the fraction.

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

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

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

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

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

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

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

$4 = 2 a + 3$

$2 = a + 3$

$b = 3$

Step 1.3.4

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

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Step 1.3.4.1

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

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

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

$2 = a + 3$

$b = 3$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

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

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Step 1.3.4.2.1.1

Simplify each term.

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Step 1.3.4.2.1.1.1

Multiply $2 (- \frac{2}{3})$ .

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Step 1.3.4.2.1.1.1.1

Multiply $- 1$ by $2$ .

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

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

$2 = a + 3$

$b = 3$

Step 1.3.4.2.1.1.1.2

Combine $- 2$ and $\frac{2}{3}$ .

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

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

$2 = a + 3$

$b = 3$

Step 1.3.4.2.1.1.1.3

Multiply $- 2$ by $2$ .

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

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

$2 = a + 3$

$b = 3$

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

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

$2 = a + 3$

$b = 3$

Step 1.3.4.2.1.1.2

Move the negative in front of the fraction.

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

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

$2 = a + 3$

$b = 3$

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

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

$2 = a + 3$

$b = 3$

Step 1.3.4.2.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

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

$2 = a + 3$

$b = 3$

Step 1.3.4.2.1.3

Combine $3$ and $\frac{3}{3}$ .

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

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

$2 = a + 3$

$b = 3$

Step 1.3.4.2.1.4

Combine the numerators over the common denominator.

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

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

$2 = a + 3$

$b = 3$

Step 1.3.4.2.1.5

Simplify the numerator.

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Step 1.3.4.2.1.5.1

Multiply $3$ by $3$ .

$4 = \frac{- 4 + 9}{3}$

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

$2 = a + 3$

$b = 3$

Step 1.3.4.2.1.5.2

Add $- 4$ and $9$ .

$4 = \frac{5}{3}$

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

$2 = a + 3$

$b = 3$

$4 = \frac{5}{3}$

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

$2 = a + 3$

$b = 3$

$4 = \frac{5}{3}$

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

$2 = a + 3$

$b = 3$

$4 = \frac{5}{3}$

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

$2 = a + 3$

$b = 3$

Step 1.3.4.3

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

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

$4 = \frac{5}{3}$

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

$b = 3$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

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

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Step 1.3.4.4.1.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

$4 = \frac{5}{3}$

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

$b = 3$

Step 1.3.4.4.1.2

Combine $3$ and $\frac{3}{3}$ .

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

$4 = \frac{5}{3}$

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

$b = 3$

Step 1.3.4.4.1.3

Combine the numerators over the common denominator.

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

$4 = \frac{5}{3}$

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

$b = 3$

Step 1.3.4.4.1.4

Simplify the numerator.

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Step 1.3.4.4.1.4.1

Multiply $3$ by $3$ .

$2 = \frac{- 2 + 9}{3}$

$4 = \frac{5}{3}$

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

$b = 3$

Step 1.3.4.4.1.4.2

Add $- 2$ and $9$ .

$2 = \frac{7}{3}$

$4 = \frac{5}{3}$

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

$b = 3$

$2 = \frac{7}{3}$

$4 = \frac{5}{3}$

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

$b = 3$

$2 = \frac{7}{3}$

$4 = \frac{5}{3}$

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

$b = 3$

$2 = \frac{7}{3}$

$4 = \frac{5}{3}$

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

$b = 3$

$2 = \frac{7}{3}$

$4 = \frac{5}{3}$

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

$b = 3$

Step 1.3.5

Since $2 = \frac{7}{3}$ 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 $3 = a \cdot 0^{2} + c$ .

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Step 2.3.1.1

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

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

$2 = a + b + c$

$4 = a \cdot 2^{2} + b (2) + c$

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

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 $0$ .

$a \cdot 0 + c = 3$

$2 = a + b + c$

$4 = a \cdot 2^{2} + b (2) + c$

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

Step 2.3.1.2.1.2

Multiply $a$ by $0$ .

$0 + c = 3$

$2 = a + b + c$

$4 = a \cdot 2^{2} + b (2) + c$

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

$0 + c = 3$

$2 = a + b + c$

$4 = a \cdot 2^{2} + b (2) + c$

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

Step 2.3.1.2.2

Add $0$ and $c$ .

$c = 3$

$2 = a + b + c$

$4 = a \cdot 2^{2} + b (2) + c$

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

$c = 3$

$2 = a + b + c$

$4 = a \cdot 2^{2} + b (2) + c$

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

$c = 3$

$2 = a + b + c$

$4 = a \cdot 2^{2} + b (2) + c$

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

Step 2.3.2

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

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Step 2.3.2.1

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

$2 = a + b + 3$

$c = 3$

$4 = a \cdot 2^{2} + b (2) + c$

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

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Remove parentheses.

$2 = a + b + 3$

$c = 3$

$4 = a \cdot 2^{2} + b (2) + c$

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

$2 = a + b + 3$

$c = 3$

$4 = a \cdot 2^{2} + b (2) + c$

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

Step 2.3.2.3

Replace all occurrences of $c$ in $4 = a \cdot 2^{2} + b (2) + c$ with $3$ .

$4 = a \cdot 2^{2} + b (2) + 3$

$2 = a + b + 3$

$c = 3$

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

Step 2.3.2.4

Simplify $4 = a \cdot 2^{2} + b (2) + 3$ .

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Step 2.3.2.4.1

Simplify the left side.

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Step 2.3.2.4.1.1

Remove parentheses.

$4 = a \cdot 2^{2} + b (2) + 3$

$2 = a + b + 3$

$c = 3$

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

$4 = a \cdot 2^{2} + b (2) + 3$

$2 = a + b + 3$

$c = 3$

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

Step 2.3.2.4.2

Simplify the right side.

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Step 2.3.2.4.2.1

Simplify each term.

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Step 2.3.2.4.2.1.1

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

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

$2 = a + b + 3$

$c = 3$

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

Step 2.3.2.4.2.1.2

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

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

$2 = a + b + 3$

$c = 3$

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

Step 2.3.2.4.2.1.3

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

$4 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

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

$4 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

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

$4 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

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

$4 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

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

Step 2.3.2.5

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

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

$4 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

Step 2.3.2.6

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

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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.

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

$4 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

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

$4 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

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 $3$ to the power of $2$ .

$1 = a \cdot 9 + b (3) + 3$

$4 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

Step 2.3.2.6.2.1.2

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

$1 = 9 \cdot a + b (3) + 3$

$4 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

Step 2.3.2.6.2.1.3

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

$1 = 9 a + 3 b + 3$

$4 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

$1 = 9 a + 3 b + 3$

$4 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

$1 = 9 a + 3 b + 3$

$4 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

$1 = 9 a + 3 b + 3$

$4 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

$1 = 9 a + 3 b + 3$

$4 = 4 a + 2 b + 3$

$2 = a + b + 3$

$c = 3$

Step 2.3.3

Solve for $a$ in $2 = a + b + 3$ .

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Step 2.3.3.1

Rewrite the equation as $a + b + 3 = 2$ .

$a + b + 3 = 2$

$1 = 9 a + 3 b + 3$

$4 = 4 a + 2 b + 3$

$c = 3$

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.

$a + 3 = 2 - b$

$1 = 9 a + 3 b + 3$

$4 = 4 a + 2 b + 3$

$c = 3$

Step 2.3.3.2.2

Subtract $3$ from both sides of the equation.

$a = 2 - b - 3$

$1 = 9 a + 3 b + 3$

$4 = 4 a + 2 b + 3$

$c = 3$

Step 2.3.3.2.3

Subtract $3$ from $2$ .

$a = - b - 1$

$1 = 9 a + 3 b + 3$

$4 = 4 a + 2 b + 3$

$c = 3$

$a = - b - 1$

$1 = 9 a + 3 b + 3$

$4 = 4 a + 2 b + 3$

$c = 3$

$a = - b - 1$

$1 = 9 a + 3 b + 3$

$4 = 4 a + 2 b + 3$

$c = 3$

Step 2.3.4

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

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Step 2.3.4.1

Replace all occurrences of $a$ in $1 = 9 a + 3 b + 3$ with $- b - 1$ .

$1 = 9 (- b - 1) + 3 b + 3$

$a = - b - 1$

$4 = 4 a + 2 b + 3$

$c = 3$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $9 (- b - 1) + 3 b + 3$ .

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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.

$1 = 9 (- b) + 9 \cdot - 1 + 3 b + 3$

$a = - b - 1$

$4 = 4 a + 2 b + 3$

$c = 3$

Step 2.3.4.2.1.1.2

Multiply $- 1$ by $9$ .

$1 = - 9 b + 9 \cdot - 1 + 3 b + 3$

$a = - b - 1$

$4 = 4 a + 2 b + 3$

$c = 3$

Step 2.3.4.2.1.1.3

Multiply $9$ by $- 1$ .

$1 = - 9 b - 9 + 3 b + 3$

$a = - b - 1$

$4 = 4 a + 2 b + 3$

$c = 3$

$1 = - 9 b - 9 + 3 b + 3$

$a = - b - 1$

$4 = 4 a + 2 b + 3$

$c = 3$

Step 2.3.4.2.1.2

Simplify by adding terms.

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Step 2.3.4.2.1.2.1

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

$1 = - 6 b - 9 + 3$

$a = - b - 1$

$4 = 4 a + 2 b + 3$

$c = 3$

Step 2.3.4.2.1.2.2

Add $- 9$ and $3$ .

$1 = - 6 b - 6$

$a = - b - 1$

$4 = 4 a + 2 b + 3$

$c = 3$

$1 = - 6 b - 6$

$a = - b - 1$

$4 = 4 a + 2 b + 3$

$c = 3$

$1 = - 6 b - 6$

$a = - b - 1$

$4 = 4 a + 2 b + 3$

$c = 3$

$1 = - 6 b - 6$

$a = - b - 1$

$4 = 4 a + 2 b + 3$

$c = 3$

Step 2.3.4.3

Replace all occurrences of $a$ in $4 = 4 a + 2 b + 3$ with $- b - 1$ .

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

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $4 (- b - 1) + 2 b + 3$ .

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Step 2.3.4.4.1.1

Simplify each term.

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Step 2.3.4.4.1.1.1

Apply the distributive property.

$4 = 4 (- b) + 4 \cdot - 1 + 2 b + 3$

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

Step 2.3.4.4.1.1.2

Multiply $- 1$ by $4$ .

$4 = - 4 b + 4 \cdot - 1 + 2 b + 3$

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

Step 2.3.4.4.1.1.3

Multiply $4$ by $- 1$ .

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

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

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

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

Step 2.3.4.4.1.2

Simplify by adding terms.

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Step 2.3.4.4.1.2.1

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

$4 = - 2 b - 4 + 3$

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

Step 2.3.4.4.1.2.2

Add $- 4$ and $3$ .

$4 = - 2 b - 1$

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

$4 = - 2 b - 1$

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

$4 = - 2 b - 1$

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

$4 = - 2 b - 1$

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

$4 = - 2 b - 1$

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

Step 2.3.5

Solve for $b$ in $4 = - 2 b - 1$ .

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Step 2.3.5.1

Rewrite the equation as $- 2 b - 1 = 4$ .

$- 2 b - 1 = 4$

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

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.

$- 2 b = 4 + 1$

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

Step 2.3.5.2.2

Add $4$ and $1$ .

$- 2 b = 5$

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

$- 2 b = 5$

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

Step 2.3.5.3

Divide each term in $- 2 b = 5$ by $- 2$ and simplify.

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Step 2.3.5.3.1

Divide each term in $- 2 b = 5$ by $- 2$ .

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

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

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.

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

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

Step 2.3.5.3.2.1.2

Divide $b$ by $1$ .

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

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

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

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

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

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

Step 2.3.5.3.3

Simplify the right side.

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Step 2.3.5.3.3.1

Move the negative in front of the fraction.

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

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

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

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

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

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

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

$1 = - 6 b - 6$

$a = - b - 1$

$c = 3$

Step 2.3.6

Replace all occurrences of $b$ with $- \frac{5}{2}$ in each equation.

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Step 2.3.6.1

Replace all occurrences of $b$ in $1 = - 6 b - 6$ with $- \frac{5}{2}$ .

$1 = - 6 (- \frac{5}{2}) - 6$

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

$a = - b - 1$

$c = 3$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $- 6 (- \frac{5}{2}) - 6$ .

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Step 2.3.6.2.1.1

Simplify each term.

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Step 2.3.6.2.1.1.1

Cancel the common factor of $2$ .

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Step 2.3.6.2.1.1.1.1

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$1 = - 6 (\frac{- 5}{2}) - 6$

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

$a = - b - 1$

$c = 3$

Step 2.3.6.2.1.1.1.2

Factor $2$ out of $- 6$ .

$1 = 2 (- 3) (\frac{- 5}{2}) - 6$

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

$a = - b - 1$

$c = 3$

Step 2.3.6.2.1.1.1.3

Cancel the common factor.

$1 = 2 \cdot (- 3 (\frac{- 5}{2})) - 6$

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

$a = - b - 1$

$c = 3$

Step 2.3.6.2.1.1.1.4

Rewrite the expression.

$1 = - 3 \cdot - 5 - 6$

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

$a = - b - 1$

$c = 3$

$1 = - 3 \cdot - 5 - 6$

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

$a = - b - 1$

$c = 3$

Step 2.3.6.2.1.1.2

Multiply $- 3$ by $- 5$ .

$1 = 15 - 6$

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

$a = - b - 1$

$c = 3$

$1 = 15 - 6$

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

$a = - b - 1$

$c = 3$

Step 2.3.6.2.1.2

Subtract $6$ from $15$ .

$1 = 9$

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

$a = - b - 1$

$c = 3$

$1 = 9$

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

$a = - b - 1$

$c = 3$

$1 = 9$

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

$a = - b - 1$

$c = 3$

Step 2.3.6.3

Replace all occurrences of $b$ in $a = - b - 1$ with $- \frac{5}{2}$ .

$a = - (- \frac{5}{2}) - 1$

$1 = 9$

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

$c = 3$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify $- (- \frac{5}{2}) - 1$ .

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Step 2.3.6.4.1.1

Multiply $- (- \frac{5}{2})$ .

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Step 2.3.6.4.1.1.1

Multiply $- 1$ by $- 1$ .

$a = 1 (\frac{5}{2}) - 1$

$1 = 9$

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

$c = 3$

Step 2.3.6.4.1.1.2

Multiply $\frac{5}{2}$ by $1$ .

$a = \frac{5}{2} - 1$

$1 = 9$

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

$c = 3$

$a = \frac{5}{2} - 1$

$1 = 9$

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

$c = 3$

Step 2.3.6.4.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = \frac{5}{2} - 1 \cdot \frac{2}{2}$

$1 = 9$

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

$c = 3$

Step 2.3.6.4.1.3

Combine $- 1$ and $\frac{2}{2}$ .

$a = \frac{5}{2} + \frac{- 1 \cdot 2}{2}$

$1 = 9$

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

$c = 3$

Step 2.3.6.4.1.4

Combine the numerators over the common denominator.

$a = \frac{5 - 1 \cdot 2}{2}$

$1 = 9$

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

$c = 3$

Step 2.3.6.4.1.5

Simplify the numerator.

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Step 2.3.6.4.1.5.1

Multiply $- 1$ by $2$ .

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

$1 = 9$

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

$c = 3$

Step 2.3.6.4.1.5.2

Subtract $2$ from $5$ .

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

$1 = 9$

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

$c = 3$

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

$1 = 9$

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

$c = 3$

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

$1 = 9$

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

$c = 3$

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

$1 = 9$

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

$c = 3$

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

$1 = 9$

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

$c = 3$

Step 2.3.7

Since $1 = 9$ 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 $x = 0$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Multiply $0$ by ${(0)}^{2}$ by adding the exponents.

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Step 2.4.1.1.1.1

Multiply $0$ by ${(0)}^{2}$ .

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Step 2.4.1.1.1.1.1

Raise $0$ to the power of $1$ .

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

Step 2.4.1.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.4.1.1.1.2

Add $1$ and $2$ .

$y = 0^{3} + (0) \cdot (0) + 0$

Step 2.4.1.1.2

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

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

Step 2.4.1.1.3

Multiply $0$ by $0$ .

$y = 0 + 0 + 0$

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, $x = 0$ . This check does not pass, since $y = 0$ and $y = 3$ . The function rule can't be quadratic.

$0 \neq 3$

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 $d$ in $3 = a \cdot 0^{3} + b \cdot 0^{2} + d$ .

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Step 3.3.1.1

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

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

$2 = a + b \cdot 0^{2} + d$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.1.2

Simplify $a \cdot 0^{3} + b \cdot 0^{2} + d$ .

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Step 3.3.1.2.1

Simplify each term.

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Step 3.3.1.2.1.1

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

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

$2 = a + b \cdot 0^{2} + d$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.1.2.1.2

Multiply $a$ by $0$ .

$0 + b \cdot 0^{2} + d = 3$

$2 = a + b \cdot 0^{2} + d$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.1.2.1.3

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

$0 + b \cdot 0 + d = 3$

$2 = a + b \cdot 0^{2} + d$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.1.2.1.4

Multiply $b$ by $0$ .

$0 + 0 + d = 3$

$2 = a + b \cdot 0^{2} + d$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

$0 + 0 + d = 3$

$2 = a + b \cdot 0^{2} + d$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.1.2.2

Combine the opposite terms in $0 + 0 + d$ .

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Step 3.3.1.2.2.1

Add $0$ and $0$ .

$0 + d = 3$

$2 = a + b \cdot 0^{2} + d$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.1.2.2.2

Add $0$ and $d$ .

$d = 3$

$2 = a + b \cdot 0^{2} + d$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

$d = 3$

$2 = a + b \cdot 0^{2} + d$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

$d = 3$

$2 = a + b \cdot 0^{2} + d$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

$d = 3$

$2 = a + b \cdot 0^{2} + d$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.2

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

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Step 3.3.2.1

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

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

$d = 3$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.2.2

Simplify $2 = a + b \cdot 0^{2} + 3$ .

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Step 3.3.2.2.1

Simplify the left side.

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Step 3.3.2.2.1.1

Remove parentheses.

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

$d = 3$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

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

$d = 3$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.2.2.2

Simplify the right side.

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Step 3.3.2.2.2.1

Simplify $a + b \cdot 0^{2} + 3$ .

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Step 3.3.2.2.2.1.1

Simplify each term.

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Step 3.3.2.2.2.1.1.1

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

$2 = a + b \cdot 0 + 3$

$d = 3$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.2.2.2.1.1.2

Multiply $b$ by $0$ .

$2 = a + 0 + 3$

$d = 3$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

$2 = a + 0 + 3$

$d = 3$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.2.2.2.1.2

Add $a$ and $0$ .

$2 = a + 3$

$d = 3$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

$2 = a + 3$

$d = 3$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

$2 = a + 3$

$d = 3$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

$2 = a + 3$

$d = 3$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + d$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.2.3

Replace all occurrences of $d$ in $4 = a \cdot 2^{3} + b \cdot 0^{2} + d$ with $3$ .

$4 = a \cdot 2^{3} + b \cdot 0^{2} + 3$

$2 = a + 3$

$d = 3$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.2.4

Simplify $4 = a \cdot 2^{3} + b \cdot 0^{2} + 3$ .

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Step 3.3.2.4.1

Simplify the left side.

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Step 3.3.2.4.1.1

Remove parentheses.

$4 = a \cdot 2^{3} + b \cdot 0^{2} + 3$

$2 = a + 3$

$d = 3$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

$4 = a \cdot 2^{3} + b \cdot 0^{2} + 3$

$2 = a + 3$

$d = 3$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.2.4.2

Simplify the right side.

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Step 3.3.2.4.2.1

Simplify $a \cdot 2^{3} + b \cdot 0^{2} + 3$ .

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Step 3.3.2.4.2.1.1

Simplify each term.

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Step 3.3.2.4.2.1.1.1

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

$4 = a \cdot 8 + b \cdot 0^{2} + 3$

$2 = a + 3$

$d = 3$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.2.4.2.1.1.2

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

$4 = 8 \cdot a + b \cdot 0^{2} + 3$

$2 = a + 3$

$d = 3$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.2.4.2.1.1.3

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

$4 = 8 a + b \cdot 0 + 3$

$2 = a + 3$

$d = 3$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.2.4.2.1.1.4

Multiply $b$ by $0$ .

$4 = 8 a + 0 + 3$

$2 = a + 3$

$d = 3$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

$4 = 8 a + 0 + 3$

$2 = a + 3$

$d = 3$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.2.4.2.1.2

Add $8 a$ and $0$ .

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + d$

Step 3.3.2.5

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

$1 = a \cdot 3^{3} + b \cdot 0^{2} + 3$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

Step 3.3.2.6

Simplify $1 = a \cdot 3^{3} + b \cdot 0^{2} + 3$ .

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Step 3.3.2.6.1

Simplify the left side.

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Step 3.3.2.6.1.1

Remove parentheses.

$1 = a \cdot 3^{3} + b \cdot 0^{2} + 3$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$1 = a \cdot 3^{3} + b \cdot 0^{2} + 3$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

Step 3.3.2.6.2

Simplify the right side.

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Step 3.3.2.6.2.1

Simplify $a \cdot 3^{3} + b \cdot 0^{2} + 3$ .

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Step 3.3.2.6.2.1.1

Simplify each term.

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Step 3.3.2.6.2.1.1.1

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

$1 = a \cdot 27 + b \cdot 0^{2} + 3$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

Step 3.3.2.6.2.1.1.2

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

$1 = 27 \cdot a + b \cdot 0^{2} + 3$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

Step 3.3.2.6.2.1.1.3

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

$1 = 27 a + b \cdot 0 + 3$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

Step 3.3.2.6.2.1.1.4

Multiply $b$ by $0$ .

$1 = 27 a + 0 + 3$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$1 = 27 a + 0 + 3$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

Step 3.3.2.6.2.1.2

Add $27 a$ and $0$ .

$1 = 27 a + 3$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$1 = 27 a + 3$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$1 = 27 a + 3$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$1 = 27 a + 3$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$1 = 27 a + 3$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

Step 3.3.3

Solve for $a$ in $1 = 27 a + 3$ .

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Step 3.3.3.1

Rewrite the equation as $27 a + 3 = 1$ .

$27 a + 3 = 1$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

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

Subtract $3$ from both sides of the equation.

$27 a = 1 - 3$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

Step 3.3.3.2.2

Subtract $3$ from $1$ .

$27 a = - 2$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$27 a = - 2$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

Step 3.3.3.3

Divide each term in $27 a = - 2$ by $27$ and simplify.

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Step 3.3.3.3.1

Divide each term in $27 a = - 2$ by $27$ .

$\frac{27 a}{27} = \frac{- 2}{27}$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

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 $27$ .

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Step 3.3.3.3.2.1.1

Cancel the common factor.

$\frac{27 a}{27} = \frac{- 2}{27}$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

Step 3.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 2}{27}$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$a = \frac{- 2}{27}$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$a = \frac{- 2}{27}$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

Step 3.3.3.3.3

Simplify the right side.

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Step 3.3.3.3.3.1

Move the negative in front of the fraction.

$a = - \frac{2}{27}$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$a = - \frac{2}{27}$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$a = - \frac{2}{27}$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

$a = - \frac{2}{27}$

$4 = 8 a + 3$

$2 = a + 3$

$d = 3$

Step 3.3.4

Replace all occurrences of $a$ with $- \frac{2}{27}$ in each equation.

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Step 3.3.4.1

Replace all occurrences of $a$ in $4 = 8 a + 3$ with $- \frac{2}{27}$ .

$4 = 8 (- \frac{2}{27}) + 3$

$a = - \frac{2}{27}$

$2 = a + 3$

$d = 3$

Step 3.3.4.2

Simplify the right side.

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Step 3.3.4.2.1

Simplify $8 (- \frac{2}{27}) + 3$ .

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Step 3.3.4.2.1.1

Simplify each term.

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Step 3.3.4.2.1.1.1

Multiply $8 (- \frac{2}{27})$ .

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Step 3.3.4.2.1.1.1.1

Multiply $- 1$ by $8$ .

$4 = - 8 (\frac{2}{27}) + 3$

$a = - \frac{2}{27}$

$2 = a + 3$

$d = 3$

Step 3.3.4.2.1.1.1.2

Combine $- 8$ and $\frac{2}{27}$ .

$4 = \frac{- 8 \cdot 2}{27} + 3$

$a = - \frac{2}{27}$

$2 = a + 3$

$d = 3$

Step 3.3.4.2.1.1.1.3

Multiply $- 8$ by $2$ .

$4 = \frac{- 16}{27} + 3$

$a = - \frac{2}{27}$

$2 = a + 3$

$d = 3$

$4 = \frac{- 16}{27} + 3$

$a = - \frac{2}{27}$

$2 = a + 3$

$d = 3$

Step 3.3.4.2.1.1.2

Move the negative in front of the fraction.

$4 = - \frac{16}{27} + 3$

$a = - \frac{2}{27}$

$2 = a + 3$

$d = 3$

$4 = - \frac{16}{27} + 3$

$a = - \frac{2}{27}$

$2 = a + 3$

$d = 3$

Step 3.3.4.2.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$4 = - \frac{16}{27} + 3 \cdot \frac{27}{27}$

$a = - \frac{2}{27}$

$2 = a + 3$

$d = 3$

Step 3.3.4.2.1.3

Combine $3$ and $\frac{27}{27}$ .

$4 = - \frac{16}{27} + \frac{3 \cdot 27}{27}$

$a = - \frac{2}{27}$

$2 = a + 3$

$d = 3$

Step 3.3.4.2.1.4

Combine the numerators over the common denominator.

$4 = \frac{- 16 + 3 \cdot 27}{27}$

$a = - \frac{2}{27}$

$2 = a + 3$

$d = 3$

Step 3.3.4.2.1.5

Simplify the numerator.

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Step 3.3.4.2.1.5.1

Multiply $3$ by $27$ .

$4 = \frac{- 16 + 81}{27}$

$a = - \frac{2}{27}$

$2 = a + 3$

$d = 3$

Step 3.3.4.2.1.5.2

Add $- 16$ and $81$ .

$4 = \frac{65}{27}$

$a = - \frac{2}{27}$

$2 = a + 3$

$d = 3$

$4 = \frac{65}{27}$

$a = - \frac{2}{27}$

$2 = a + 3$

$d = 3$

$4 = \frac{65}{27}$

$a = - \frac{2}{27}$

$2 = a + 3$

$d = 3$

$4 = \frac{65}{27}$

$a = - \frac{2}{27}$

$2 = a + 3$

$d = 3$

Step 3.3.4.3

Replace all occurrences of $a$ in $2 = a + 3$ with $- \frac{2}{27}$ .

$2 = (- \frac{2}{27}) + 3$

$4 = \frac{65}{27}$

$a = - \frac{2}{27}$

$d = 3$

Step 3.3.4.4

Simplify the right side.

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Step 3.3.4.4.1

Simplify $(- \frac{2}{27}) + 3$ .

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Step 3.3.4.4.1.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$2 = - \frac{2}{27} + 3 \cdot \frac{27}{27}$

$4 = \frac{65}{27}$

$a = - \frac{2}{27}$

$d = 3$

Step 3.3.4.4.1.2

Combine $3$ and $\frac{27}{27}$ .

$2 = - \frac{2}{27} + \frac{3 \cdot 27}{27}$

$4 = \frac{65}{27}$

$a = - \frac{2}{27}$

$d = 3$

Step 3.3.4.4.1.3

Combine the numerators over the common denominator.

$2 = \frac{- 2 + 3 \cdot 27}{27}$

$4 = \frac{65}{27}$

$a = - \frac{2}{27}$

$d = 3$

Step 3.3.4.4.1.4

Simplify the numerator.

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Step 3.3.4.4.1.4.1

Multiply $3$ by $27$ .

$2 = \frac{- 2 + 81}{27}$

$4 = \frac{65}{27}$

$a = - \frac{2}{27}$

$d = 3$

Step 3.3.4.4.1.4.2

Add $- 2$ and $81$ .

$2 = \frac{79}{27}$

$4 = \frac{65}{27}$

$a = - \frac{2}{27}$

$d = 3$

$2 = \frac{79}{27}$

$4 = \frac{65}{27}$

$a = - \frac{2}{27}$

$d = 3$

$2 = \frac{79}{27}$

$4 = \frac{65}{27}$

$a = - \frac{2}{27}$

$d = 3$

$2 = \frac{79}{27}$

$4 = \frac{65}{27}$

$a = - \frac{2}{27}$

$d = 3$

$2 = \frac{79}{27}$

$4 = \frac{65}{27}$

$a = - \frac{2}{27}$

$d = 3$

Step 3.3.5

Since $2 = \frac{79}{27}$ 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 $x = 0$ .

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Step 3.4.1.1

Simplify each term.

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Step 3.4.1.1.1

Multiply $0$ by ${(0)}^{3}$ by adding the exponents.

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Step 3.4.1.1.1.1

Multiply $0$ by ${(0)}^{3}$ .

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Step 3.4.1.1.1.1.1

Raise $0$ to the power of $1$ .

$y = 0 \cdot {(0)}^{3} + (0) \cdot (0^{2}) + (0) \cdot ((0) \cdot 0)$

Step 3.4.1.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = 0^{1 + 3} + (0) \cdot (0^{2}) + (0) \cdot ((0) \cdot 0)$

Step 3.4.1.1.1.2

Add $1$ and $3$ .

$y = 0^{4} + (0) \cdot (0^{2}) + (0) \cdot ((0) \cdot 0)$

Step 3.4.1.1.2

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

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

Step 3.4.1.1.3

Multiply $0$ by $0^{2}$ by adding the exponents.

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Step 3.4.1.1.3.1

Multiply $0$ by $0^{2}$ .

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Step 3.4.1.1.3.1.1

Raise $0$ to the power of $1$ .

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

Step 3.4.1.1.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.4.1.1.3.2

Add $1$ and $2$ .

$y = 0 + 0^{3} + (0) \cdot ((0) \cdot 0)$

Step 3.4.1.1.4

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

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

Step 3.4.1.1.5

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

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Step 3.4.1.1.5.1

Multiply $0$ 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, $x = 0$ . This check does not pass, since $y = 0$ and $y = 3$ . The function rule can't be cubic.

$0 \neq 3$

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.