Algebra Examples

$\begin{array}{c} x & y \\ 0.5 & - 0.75 \\ 1 & 0 \\ 1.5 & 0.5 \end{array}$

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 $y = a x + b$ .

$- 0.75 = a (0.5) + b 0 = a (1) + b 0.5 = a (1.5) + b$

Step 1.3

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

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Step 1.3.1

Solve for $a$ in $0 = a + b$ .

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Step 1.3.1.1

Rewrite the equation as $a + b = 0$ .

$a + b = 0$

$- 0.75 = a (0.5) + b$

$0.5 = a (1.5) + b$

Step 1.3.1.2

Subtract $b$ from both sides of the equation.

$a = - b$

$- 0.75 = a (0.5) + b$

$0.5 = a (1.5) + b$

$a = - b$

$- 0.75 = a (0.5) + b$

$0.5 = a (1.5) + b$

Step 1.3.2

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

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Step 1.3.2.1

Replace all occurrences of $a$ in $- 0.75 = a (0.5) + b$ with $- b$ .

$- 0.75 = (- b) (0.5) + b$

$a = - b$

$0.5 = a (1.5) + b$

Step 1.3.2.2

Simplify the right side.

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Step 1.3.2.2.1

Simplify $(- b) (0.5) + b$ .

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Step 1.3.2.2.1.1

Multiply $0.5$ by $- 1$ .

$- 0.75 = - 0.5 b + b$

$a = - b$

$0.5 = a (1.5) + b$

Step 1.3.2.2.1.2

Add $- 0.5 b$ and $b$ .

$- 0.75 = 0.5 b$

$a = - b$

$0.5 = a (1.5) + b$

$- 0.75 = 0.5 b$

$a = - b$

$0.5 = a (1.5) + b$

$- 0.75 = 0.5 b$

$a = - b$

$0.5 = a (1.5) + b$

Step 1.3.2.3

Replace all occurrences of $a$ in $0.5 = a (1.5) + b$ with $- b$ .

$0.5 = (- b) (1.5) + b$

$- 0.75 = 0.5 b$

$a = - b$

Step 1.3.2.4

Simplify the right side.

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Step 1.3.2.4.1

Simplify $(- b) (1.5) + b$ .

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Step 1.3.2.4.1.1

Multiply $1.5$ by $- 1$ .

$0.5 = - 1.5 b + b$

$- 0.75 = 0.5 b$

$a = - b$

Step 1.3.2.4.1.2

Add $- 1.5 b$ and $b$ .

$0.5 = - 0.5 b$

$- 0.75 = 0.5 b$

$a = - b$

$0.5 = - 0.5 b$

$- 0.75 = 0.5 b$

$a = - b$

$0.5 = - 0.5 b$

$- 0.75 = 0.5 b$

$a = - b$

$0.5 = - 0.5 b$

$- 0.75 = 0.5 b$

$a = - b$

Step 1.3.3

Solve for $b$ in $0.5 = - 0.5 b$ .

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Step 1.3.3.1

Rewrite the equation as $- 0.5 b = 0.5$ .

$- 0.5 b = 0.5$

$- 0.75 = 0.5 b$

$a = - b$

Step 1.3.3.2

Divide each term in $- 0.5 b = 0.5$ by $- 0.5$ and simplify.

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Step 1.3.3.2.1

Divide each term in $- 0.5 b = 0.5$ by $- 0.5$ .

$\frac{- 0.5 b}{- 0.5} = \frac{0.5}{- 0.5}$

$- 0.75 = 0.5 b$

$a = - b$

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

Cancel the common factor of $- 0.5$ .

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Step 1.3.3.2.2.1.1

Cancel the common factor.

$\frac{- 0.5 b}{- 0.5} = \frac{0.5}{- 0.5}$

$- 0.75 = 0.5 b$

$a = - b$

Step 1.3.3.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0.5}{- 0.5}$

$- 0.75 = 0.5 b$

$a = - b$

$b = \frac{0.5}{- 0.5}$

$- 0.75 = 0.5 b$

$a = - b$

$b = \frac{0.5}{- 0.5}$

$- 0.75 = 0.5 b$

$a = - b$

Step 1.3.3.2.3

Simplify the right side.

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Step 1.3.3.2.3.1

Divide $0.5$ by $- 0.5$ .

$b = - 1$

$- 0.75 = 0.5 b$

$a = - b$

$b = - 1$

$- 0.75 = 0.5 b$

$a = - b$

$b = - 1$

$- 0.75 = 0.5 b$

$a = - b$

$b = - 1$

$- 0.75 = 0.5 b$

$a = - b$

Step 1.3.4

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

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Step 1.3.4.1

Replace all occurrences of $b$ in $- 0.75 = 0.5 b$ with $- 1$ .

$- 0.75 = 0.5 (- 1)$

$b = - 1$

$a = - b$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Multiply $0.5$ by $- 1$ .

$- 0.75 = - 0.5$

$b = - 1$

$a = - b$

$- 0.75 = - 0.5$

$b = - 1$

$a = - b$

Step 1.3.4.3

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

$a = - (- 1)$

$- 0.75 = - 0.5$

$b = - 1$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Multiply $- 1$ by $- 1$ .

$a = 1$

$- 0.75 = - 0.5$

$b = - 1$

$a = 1$

$- 0.75 = - 0.5$

$b = - 1$

$a = 1$

$- 0.75 = - 0.5$

$b = - 1$

Step 1.3.5

Since $- 0.75 = - 0.5$ 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 $a$ in $0 = a + b + c$ .

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Step 2.3.1.1

Rewrite the equation as $a + b + c = 0$ .

$a + b + c = 0$

$- 0.75 = a \cdot {0.5}^{2} + b (0.5) + c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Subtract $b$ from both sides of the equation.

$a + c = - b$

$- 0.75 = a \cdot {0.5}^{2} + b (0.5) + c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

Step 2.3.1.2.2

Subtract $c$ from both sides of the equation.

$a = - b - c$

$- 0.75 = a \cdot {0.5}^{2} + b (0.5) + c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

$a = - b - c$

$- 0.75 = a \cdot {0.5}^{2} + b (0.5) + c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

$a = - b - c$

$- 0.75 = a \cdot {0.5}^{2} + b (0.5) + c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

Step 2.3.2

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

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Step 2.3.2.1

Replace all occurrences of $a$ in $- 0.75 = a \cdot {0.5}^{2} + b (0.5) + c$ with $- b - c$ .

$- 0.75 = (- b - c) \cdot {0.5}^{2} + b (0.5) + c$

$a = - b - c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

Step 2.3.2.2

Simplify the right side.

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Step 2.3.2.2.1

Simplify $(- b - c) \cdot {0.5}^{2} + b (0.5) + c$ .

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Step 2.3.2.2.1.1

Simplify each term.

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Step 2.3.2.2.1.1.1

Raise $0.5$ to the power of $2$ .

$- 0.75 = (- b - c) \cdot 0.25 + b (0.5) + c$

$a = - b - c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

Step 2.3.2.2.1.1.2

Apply the distributive property.

$- 0.75 = - b \cdot 0.25 - c \cdot 0.25 + b (0.5) + c$

$a = - b - c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

Step 2.3.2.2.1.1.3

Multiply $0.25$ by $- 1$ .

$- 0.75 = - 0.25 b - c \cdot 0.25 + b (0.5) + c$

$a = - b - c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

Step 2.3.2.2.1.1.4

Multiply $0.25$ by $- 1$ .

$- 0.75 = - 0.25 b - 0.25 c + b (0.5) + c$

$a = - b - c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

Step 2.3.2.2.1.1.5

Move $0.5$ to the left of $b$ .

$- 0.75 = - 0.25 b - 0.25 c + 0.5 b + c$

$a = - b - c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

$- 0.75 = - 0.25 b - 0.25 c + 0.5 b + c$

$a = - b - c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

Step 2.3.2.2.1.2

Simplify by adding terms.

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Step 2.3.2.2.1.2.1

Add $- 0.25 b$ and $0.5 b$ .

$- 0.75 = 0.25 b - 0.25 c + c$

$a = - b - c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

Step 2.3.2.2.1.2.2

Add $- 0.25 c$ and $c$ .

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

$0.5 = a \cdot {1.5}^{2} + b (1.5) + c$

Step 2.3.2.3

Replace all occurrences of $a$ in $0.5 = a \cdot {1.5}^{2} + b (1.5) + c$ with $- b - c$ .

$0.5 = (- b - c) \cdot {1.5}^{2} + b (1.5) + c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.2.4

Simplify the right side.

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Step 2.3.2.4.1

Simplify $(- b - c) \cdot {1.5}^{2} + b (1.5) + c$ .

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Step 2.3.2.4.1.1

Simplify each term.

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Step 2.3.2.4.1.1.1

Raise $1.5$ to the power of $2$ .

$0.5 = (- b - c) \cdot 2.25 + b (1.5) + c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.2.4.1.1.2

Apply the distributive property.

$0.5 = - b \cdot 2.25 - c \cdot 2.25 + b (1.5) + c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.2.4.1.1.3

Multiply $2.25$ by $- 1$ .

$0.5 = - 2.25 b - c \cdot 2.25 + b (1.5) + c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.2.4.1.1.4

Multiply $2.25$ by $- 1$ .

$0.5 = - 2.25 b - 2.25 c + b (1.5) + c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.2.4.1.1.5

Move $1.5$ to the left of $b$ .

$0.5 = - 2.25 b - 2.25 c + 1.5 b + c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

$0.5 = - 2.25 b - 2.25 c + 1.5 b + c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.2.4.1.2

Simplify by adding terms.

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Step 2.3.2.4.1.2.1

Add $- 2.25 b$ and $1.5 b$ .

$0.5 = - 0.75 b - 2.25 c + c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.2.4.1.2.2

Add $- 2.25 c$ and $c$ .

$0.5 = - 0.75 b - 1.25 c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

$0.5 = - 0.75 b - 1.25 c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

$0.5 = - 0.75 b - 1.25 c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

$0.5 = - 0.75 b - 1.25 c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

$0.5 = - 0.75 b - 1.25 c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.3

Solve for $b$ in $0.5 = - 0.75 b - 1.25 c$ .

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Step 2.3.3.1

Rewrite the equation as $- 0.75 b - 1.25 c = 0.5$ .

$- 0.75 b - 1.25 c = 0.5$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.3.2

Add $1.25 c$ to both sides of the equation.

$- 0.75 b = 0.5 + 1.25 c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.3.3

Divide each term in $- 0.75 b = 0.5 + 1.25 c$ by $- 0.75$ and simplify.

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Step 2.3.3.3.1

Divide each term in $- 0.75 b = 0.5 + 1.25 c$ by $- 0.75$ .

$\frac{- 0.75 b}{- 0.75} = \frac{0.5}{- 0.75} + \frac{1.25 c}{- 0.75}$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.3.3.2

Simplify the left side.

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Step 2.3.3.3.2.1

Cancel the common factor of $- 0.75$ .

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Step 2.3.3.3.2.1.1

Cancel the common factor.

$\frac{- 0.75 b}{- 0.75} = \frac{0.5}{- 0.75} + \frac{1.25 c}{- 0.75}$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.3.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{0.5}{- 0.75} + \frac{1.25 c}{- 0.75}$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

$b = \frac{0.5}{- 0.75} + \frac{1.25 c}{- 0.75}$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

$b = \frac{0.5}{- 0.75} + \frac{1.25 c}{- 0.75}$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.3.3.3

Simplify the right side.

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Step 2.3.3.3.3.1

Simplify each term.

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Step 2.3.3.3.3.1.1

Divide $0.5$ by $- 0.75$ .

$b = - 0.\overline{6} + \frac{1.25 c}{- 0.75}$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.3.3.3.1.2

Move the negative in front of the fraction.

$b = - 0.\overline{6} - \frac{1.25 c}{0.75}$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.3.3.3.1.3

Factor $1.25$ out of $1.25 c$ .

$b = - 0.\overline{6} - \frac{1.25 (c)}{0.75}$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.3.3.3.1.4

Factor $0.75$ out of $0.75$ .

$b = - 0.\overline{6} - \frac{1.25 (c)}{0.75 (1)}$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.3.3.3.1.5

Separate fractions.

$b = - 0.\overline{6} - (\frac{1.25}{0.75} \cdot \frac{c}{1})$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.3.3.3.1.6

Divide $1.25$ by $0.75$ .

$b = - 0.\overline{6} - (1.\overline{6} (\frac{c}{1}))$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.3.3.3.1.7

Divide $c$ by $1$ .

$b = - 0.\overline{6} - (1.\overline{6} c)$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.3.3.3.1.8

Multiply $1.\overline{6}$ by $- 1$ .

$b = - 0.\overline{6} - 1.\overline{6} c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$- 0.75 = 0.25 b + 0.75 c$

$a = - b - c$

Step 2.3.4

Replace all occurrences of $b$ with $- 0.\overline{6} - 1.\overline{6} c$ in each equation.

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Step 2.3.4.1

Replace all occurrences of $b$ in $- 0.75 = 0.25 b + 0.75 c$ with $- 0.\overline{6} - 1.\overline{6} c$ .

$- 0.75 = 0.25 (- 0.\overline{6} - 1.\overline{6} c) + 0.75 c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$a = - b - c$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $0.25 (- 0.\overline{6} - 1.\overline{6} c) + 0.75 c$ .

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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.

$- 0.75 = 0.25 \cdot - 0.\overline{6} + 0.25 (- 1.\overline{6} c) + 0.75 c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$a = - b - c$

Step 2.3.4.2.1.1.2

Multiply $0.25$ by $- 0.\overline{6}$ .

$- 0.75 = - 0.1\overline{6} + 0.25 (- 1.\overline{6} c) + 0.75 c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$a = - b - c$

Step 2.3.4.2.1.1.3

Multiply $- 1.\overline{6}$ by $0.25$ .

$- 0.75 = - 0.1\overline{6} - 0.41\overline{6} c + 0.75 c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$a = - b - c$

$- 0.75 = - 0.1\overline{6} - 0.41\overline{6} c + 0.75 c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$a = - b - c$

Step 2.3.4.2.1.2

Add $- 0.41\overline{6} c$ and $0.75 c$ .

$- 0.75 = - 0.1\overline{6} + 0.\overline{3} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$a = - b - c$

$- 0.75 = - 0.1\overline{6} + 0.\overline{3} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$a = - b - c$

$- 0.75 = - 0.1\overline{6} + 0.\overline{3} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$a = - b - c$

Step 2.3.4.3

Replace all occurrences of $b$ in $a = - b - c$ with $- 0.\overline{6} - 1.\overline{6} c$ .

$a = - (- 0.\overline{6} - 1.\overline{6} c) - c$

$- 0.75 = - 0.1\overline{6} + 0.\overline{3} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $- (- 0.\overline{6} - 1.\overline{6} c) - c$ .

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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.

$a = 0.\overline{6} - (- 1.\overline{6} c) - c$

$- 0.75 = - 0.1\overline{6} + 0.\overline{3} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

Step 2.3.4.4.1.1.2

Multiply $- 1$ by $- 0.\overline{6}$ .

$a = 0.\overline{6} - (- 1.\overline{6} c) - c$

$- 0.75 = - 0.1\overline{6} + 0.\overline{3} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

Step 2.3.4.4.1.1.3

Multiply $- 1.\overline{6}$ by $- 1$ .

$a = 0.\overline{6} + 1.\overline{6} c - c$

$- 0.75 = - 0.1\overline{6} + 0.\overline{3} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$a = 0.\overline{6} + 1.\overline{6} c - c$

$- 0.75 = - 0.1\overline{6} + 0.\overline{3} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

Step 2.3.4.4.1.2

Subtract $c$ from $1.\overline{6} c$ .

$a = 0.\overline{6} + 0.\overline{6} c$

$- 0.75 = - 0.1\overline{6} + 0.\overline{3} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$a = 0.\overline{6} + 0.\overline{6} c$

$- 0.75 = - 0.1\overline{6} + 0.\overline{3} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$a = 0.\overline{6} + 0.\overline{6} c$

$- 0.75 = - 0.1\overline{6} + 0.\overline{3} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$a = 0.\overline{6} + 0.\overline{6} c$

$- 0.75 = - 0.1\overline{6} + 0.\overline{3} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

Step 2.3.5

Solve for $c$ in $- 0.75 = - 0.1\overline{6} + 0.\overline{3} c$ .

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Step 2.3.5.1

Rewrite the equation as $- 0.1\overline{6} + 0.\overline{3} c = - 0.75$ .

$- 0.1\overline{6} + 0.\overline{3} c = - 0.75$

$a = 0.\overline{6} + 0.\overline{6} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

Step 2.3.5.2

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

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Step 2.3.5.2.1

Add $0.1\overline{6}$ to both sides of the equation.

$0.\overline{3} c = - 0.75 + 0.1\overline{6}$

$a = 0.\overline{6} + 0.\overline{6} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

Step 2.3.5.2.2

Add $- 0.75$ and $0.1\overline{6}$ .

$0.\overline{3} c = - 0.58\overline{3}$

$a = 0.\overline{6} + 0.\overline{6} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$0.\overline{3} c = - 0.58\overline{3}$

$a = 0.\overline{6} + 0.\overline{6} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

Step 2.3.5.3

Divide each term in $0.\overline{3} c = - 0.58\overline{3}$ by $0.\overline{3}$ and simplify.

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Step 2.3.5.3.1

Divide each term in $0.\overline{3} c = - 0.58\overline{3}$ by $0.\overline{3}$ .

$\frac{0.\overline{3} c}{0.\overline{3}} = \frac{- 0.58\overline{3}}{0.\overline{3}}$

$a = 0.\overline{6} + 0.\overline{6} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

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 $0.\overline{3}$ .

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Step 2.3.5.3.2.1.1

Cancel the common factor.

$\frac{0.\overline{3} c}{0.\overline{3}} = \frac{- 0.58\overline{3}}{0.\overline{3}}$

$a = 0.\overline{6} + 0.\overline{6} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

Step 2.3.5.3.2.1.2

Divide $c$ by $1$ .

$c = \frac{- 0.58\overline{3}}{0.\overline{3}}$

$a = 0.\overline{6} + 0.\overline{6} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$c = \frac{- 0.58\overline{3}}{0.\overline{3}}$

$a = 0.\overline{6} + 0.\overline{6} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$c = \frac{- 0.58\overline{3}}{0.\overline{3}}$

$a = 0.\overline{6} + 0.\overline{6} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

Step 2.3.5.3.3

Simplify the right side.

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Step 2.3.5.3.3.1

Divide $- 0.58\overline{3}$ by $0.\overline{3}$ .

$c = - 1.75$

$a = 0.\overline{6} + 0.\overline{6} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$c = - 1.75$

$a = 0.\overline{6} + 0.\overline{6} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$c = - 1.75$

$a = 0.\overline{6} + 0.\overline{6} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

$c = - 1.75$

$a = 0.\overline{6} + 0.\overline{6} c$

$b = - 0.\overline{6} - 1.\overline{6} c$

Step 2.3.6

Replace all occurrences of $c$ with $- 1.75$ in each equation.

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Step 2.3.6.1

Replace all occurrences of $c$ in $a = 0.\overline{6} + 0.\overline{6} c$ with $- 1.75$ .

$a = 0.\overline{6} + 0.\overline{6} (- 1.75)$

$c = - 1.75$

$b = - 0.\overline{6} - 1.\overline{6} c$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $0.\overline{6} + 0.\overline{6} (- 1.75)$ .

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Step 2.3.6.2.1.1

Multiply $0.\overline{6}$ by $- 1.75$ .

$a = 0.\overline{6} - 1.1\overline{6}$

$c = - 1.75$

$b = - 0.\overline{6} - 1.\overline{6} c$

Step 2.3.6.2.1.2

Subtract $1.1\overline{6}$ from $0.\overline{6}$ .

$a = - 0.5$

$c = - 1.75$

$b = - 0.\overline{6} - 1.\overline{6} c$

$a = - 0.5$

$c = - 1.75$

$b = - 0.\overline{6} - 1.\overline{6} c$

$a = - 0.5$

$c = - 1.75$

$b = - 0.\overline{6} - 1.\overline{6} c$

Step 2.3.6.3

Replace all occurrences of $c$ in $b = - 0.\overline{6} - 1.\overline{6} c$ with $- 1.75$ .

$b = - 0.\overline{6} - 1.\overline{6} \cdot - 1.75$

$a = - 0.5$

$c = - 1.75$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify $- 0.\overline{6} - 1.\overline{6} \cdot - 1.75$ .

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Step 2.3.6.4.1.1

Multiply $- 1.\overline{6}$ by $- 1.75$ .

$b = - 0.\overline{6} + 2.91\overline{6}$

$a = - 0.5$

$c = - 1.75$

Step 2.3.6.4.1.2

Add $- 0.\overline{6}$ and $2.91\overline{6}$ .

$b = 2.25$

$a = - 0.5$

$c = - 1.75$

$b = 2.25$

$a = - 0.5$

$c = - 1.75$

$b = 2.25$

$a = - 0.5$

$c = - 1.75$

$b = 2.25$

$a = - 0.5$

$c = - 1.75$

Step 2.3.7

List all of the solutions.

$b = 2.25, a = - 0.5, c = - 1.75$

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 $a = - 0.5$ , $b = 2.25$ , $c = - 1.75$ , and $x = 0.5$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Raise $0.5$ to the power of $2$ .

$y = - 0.5 \cdot 0.25 + (2.25) \cdot (0.5) - 1.75$

Step 2.4.1.1.2

Multiply $- 0.5$ by $0.25$ .

$y = - 0.125 + (2.25) \cdot (0.5) - 1.75$

Step 2.4.1.1.3

Multiply $2.25$ by $0.5$ .

$y = - 0.125 + 1.125 - 1.75$

Step 2.4.1.2

Simplify by adding and subtracting.

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Step 2.4.1.2.1

Add $- 0.125$ and $1.125$ .

$y = 1 - 1.75$

Step 2.4.1.2.2

Subtract $1.75$ from $1$ .

$y = - 0.75$

Step 2.4.2

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 0.5$ . This check does not pass, since $y = - 0.75$ and $y = - 0.75$ . The function rule can't be quadratic.

$- 0.75 \neq - 0.75$

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

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

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