Algebra Examples

$\begin{array}{c} x & y \\ - 1 & \frac{8}{7} \\ 0 & 8 \\ 1 & 56 \end{array}$

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 $y = a x + b$ .

$\frac{8}{7} = a (- 1) + b 8 = a (0) + b 56 = a (1) + b$

Step 1.3

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

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Step 1.3.1

Rewrite the equation as $b = 8$ .

$b = 8$

$\frac{8}{7} = a (- 1) + b$

$56 = a + b$

Step 1.3.2

Replace all occurrences of $b$ with $8$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $\frac{8}{7} = a (- 1) + b$ with $8$ .

$\frac{8}{7} = a (- 1) + 8$

$b = 8$

$56 = a + b$

Step 1.3.2.2

Simplify $\frac{8}{7} = a (- 1) + 8$ .

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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.

$\frac{8}{7} = a (- 1) + 8$

$b = 8$

$56 = a + b$

$\frac{8}{7} = a (- 1) + 8$

$b = 8$

$56 = a + b$

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 $a$ .

$\frac{8}{7} = - 1 a + 8$

$b = 8$

$56 = a + b$

Step 1.3.2.2.2.1.2

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

$\frac{8}{7} = - a + 8$

$b = 8$

$56 = a + b$

$\frac{8}{7} = - a + 8$

$b = 8$

$56 = a + b$

$\frac{8}{7} = - a + 8$

$b = 8$

$56 = a + b$

$\frac{8}{7} = - a + 8$

$b = 8$

$56 = a + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $56 = a + b$ with $8$ .

$56 = a + 8$

$\frac{8}{7} = - a + 8$

$b = 8$

Step 1.3.2.4

Simplify the left side.

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Step 1.3.2.4.1

Remove parentheses.

$56 = a + 8$

$\frac{8}{7} = - a + 8$

$b = 8$

$56 = a + 8$

$\frac{8}{7} = - a + 8$

$b = 8$

$56 = a + 8$

$\frac{8}{7} = - a + 8$

$b = 8$

Step 1.3.3

Solve for $a$ in $56 = a + 8$ .

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Step 1.3.3.1

Rewrite the equation as $a + 8 = 56$ .

$a + 8 = 56$

$\frac{8}{7} = - a + 8$

$b = 8$

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 $8$ from both sides of the equation.

$a = 56 - 8$

$\frac{8}{7} = - a + 8$

$b = 8$

Step 1.3.3.2.2

Subtract $8$ from $56$ .

$a = 48$

$\frac{8}{7} = - a + 8$

$b = 8$

$a = 48$

$\frac{8}{7} = - a + 8$

$b = 8$

$a = 48$

$\frac{8}{7} = - a + 8$

$b = 8$

Step 1.3.4

Replace all occurrences of $a$ with $48$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $a$ in $\frac{8}{7} = - a + 8$ with $48$ .

$\frac{8}{7} = - (48) + 8$

$a = 48$

$b = 8$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $- (48) + 8$ .

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Step 1.3.4.2.1.1

Multiply $- 1$ by $48$ .

$\frac{8}{7} = - 48 + 8$

$a = 48$

$b = 8$

Step 1.3.4.2.1.2

Add $- 48$ and $8$ .

$\frac{8}{7} = - 40$

$a = 48$

$b = 8$

$\frac{8}{7} = - 40$

$a = 48$

$b = 8$

$\frac{8}{7} = - 40$

$a = 48$

$b = 8$

$\frac{8}{7} = - 40$

$a = 48$

$b = 8$

Step 1.3.5

Since $\frac{8}{7} = - 40$ 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 $8 = a \cdot 0^{2} + c$ .

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Step 2.3.1.1

Rewrite the equation as $a \cdot 0^{2} + c = 8$ .

$a \cdot 0^{2} + c = 8$

$\frac{8}{7} = a {(- 1)}^{2} + b (- 1) + c$

$56 = a + b + c$

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 $0$ .

$a \cdot 0 + c = 8$

$\frac{8}{7} = a {(- 1)}^{2} + b (- 1) + c$

$56 = a + b + c$

Step 2.3.1.2.1.2

Multiply $a$ by $0$ .

$0 + c = 8$

$\frac{8}{7} = a {(- 1)}^{2} + b (- 1) + c$

$56 = a + b + c$

$0 + c = 8$

$\frac{8}{7} = a {(- 1)}^{2} + b (- 1) + c$

$56 = a + b + c$

Step 2.3.1.2.2

Add $0$ and $c$ .

$c = 8$

$\frac{8}{7} = a {(- 1)}^{2} + b (- 1) + c$

$56 = a + b + c$

$c = 8$

$\frac{8}{7} = a {(- 1)}^{2} + b (- 1) + c$

$56 = a + b + c$

$c = 8$

$\frac{8}{7} = a {(- 1)}^{2} + b (- 1) + c$

$56 = a + b + c$

Step 2.3.2

Replace all occurrences of $c$ with $8$ in each equation.

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Step 2.3.2.1

Replace all occurrences of $c$ in $\frac{8}{7} = a {(- 1)}^{2} + b (- 1) + c$ with $8$ .

$\frac{8}{7} = a {(- 1)}^{2} + b (- 1) + 8$

$c = 8$

$56 = a + b + c$

Step 2.3.2.2

Simplify $\frac{8}{7} = a {(- 1)}^{2} + b (- 1) + 8$ .

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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.

$\frac{8}{7} = a {(- 1)}^{2} + b (- 1) + 8$

$c = 8$

$56 = a + b + c$

$\frac{8}{7} = a {(- 1)}^{2} + b (- 1) + 8$

$c = 8$

$56 = a + b + c$

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 $2$ .

$\frac{8}{7} = a \cdot 1 + b (- 1) + 8$

$c = 8$

$56 = a + b + c$

Step 2.3.2.2.2.1.2

Multiply $a$ by $1$ .

$\frac{8}{7} = a + b (- 1) + 8$

$c = 8$

$56 = a + b + c$

Step 2.3.2.2.2.1.3

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

$\frac{8}{7} = a - 1 b + 8$

$c = 8$

$56 = a + b + c$

Step 2.3.2.2.2.1.4

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

$\frac{8}{7} = a - b + 8$

$c = 8$

$56 = a + b + c$

$\frac{8}{7} = a - b + 8$

$c = 8$

$56 = a + b + c$

$\frac{8}{7} = a - b + 8$

$c = 8$

$56 = a + b + c$

$\frac{8}{7} = a - b + 8$

$c = 8$

$56 = a + b + c$

Step 2.3.2.3

Replace all occurrences of $c$ in $56 = a + b + c$ with $8$ .

$56 = a + b + 8$

$\frac{8}{7} = a - b + 8$

$c = 8$

Step 2.3.2.4

Simplify the left side.

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Step 2.3.2.4.1

Remove parentheses.

$56 = a + b + 8$

$\frac{8}{7} = a - b + 8$

$c = 8$

$56 = a + b + 8$

$\frac{8}{7} = a - b + 8$

$c = 8$

$56 = a + b + 8$

$\frac{8}{7} = a - b + 8$

$c = 8$

Step 2.3.3

Solve for $a$ in $56 = a + b + 8$ .

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Step 2.3.3.1

Rewrite the equation as $a + b + 8 = 56$ .

$a + b + 8 = 56$

$\frac{8}{7} = a - b + 8$

$c = 8$

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.

$a + 8 = 56 - b$

$\frac{8}{7} = a - b + 8$

$c = 8$

Step 2.3.3.2.2

Subtract $8$ from both sides of the equation.

$a = 56 - b - 8$

$\frac{8}{7} = a - b + 8$

$c = 8$

Step 2.3.3.2.3

Subtract $8$ from $56$ .

$a = - b + 48$

$\frac{8}{7} = a - b + 8$

$c = 8$

$a = - b + 48$

$\frac{8}{7} = a - b + 8$

$c = 8$

$a = - b + 48$

$\frac{8}{7} = a - b + 8$

$c = 8$

Step 2.3.4

Replace all occurrences of $a$ with $- b + 48$ in each equation.

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Step 2.3.4.1

Replace all occurrences of $a$ in $\frac{8}{7} = a - b + 8$ with $- b + 48$ .

$\frac{8}{7} = (- b + 48) - b + 8$

$a = - b + 48$

$c = 8$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $(- b + 48) - b + 8$ .

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Step 2.3.4.2.1.1

Subtract $b$ from $- b$ .

$\frac{8}{7} = - 2 b + 48 + 8$

$a = - b + 48$

$c = 8$

Step 2.3.4.2.1.2

Add $48$ and $8$ .

$\frac{8}{7} = - 2 b + 56$

$a = - b + 48$

$c = 8$

$\frac{8}{7} = - 2 b + 56$

$a = - b + 48$

$c = 8$

$\frac{8}{7} = - 2 b + 56$

$a = - b + 48$

$c = 8$

$\frac{8}{7} = - 2 b + 56$

$a = - b + 48$

$c = 8$

Step 2.3.5

Solve for $b$ in $\frac{8}{7} = - 2 b + 56$ .

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Step 2.3.5.1

Rewrite the equation as $- 2 b + 56 = \frac{8}{7}$ .

$- 2 b + 56 = \frac{8}{7}$

$a = - b + 48$

$c = 8$

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

Subtract $56$ from both sides of the equation.

$- 2 b = \frac{8}{7} - 56$

$a = - b + 48$

$c = 8$

Step 2.3.5.2.2

To write $- 56$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- 2 b = \frac{8}{7} - 56 \cdot \frac{7}{7}$

$a = - b + 48$

$c = 8$

Step 2.3.5.2.3

Combine $- 56$ and $\frac{7}{7}$ .

$- 2 b = \frac{8}{7} + \frac{- 56 \cdot 7}{7}$

$a = - b + 48$

$c = 8$

Step 2.3.5.2.4

Combine the numerators over the common denominator.

$- 2 b = \frac{8 - 56 \cdot 7}{7}$

$a = - b + 48$

$c = 8$

Step 2.3.5.2.5

Simplify the numerator.

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Step 2.3.5.2.5.1

Multiply $- 56$ by $7$ .

$- 2 b = \frac{8 - 392}{7}$

$a = - b + 48$

$c = 8$

Step 2.3.5.2.5.2

Subtract $392$ from $8$ .

$- 2 b = \frac{- 384}{7}$

$a = - b + 48$

$c = 8$

$- 2 b = \frac{- 384}{7}$

$a = - b + 48$

$c = 8$

Step 2.3.5.2.6

Move the negative in front of the fraction.

$- 2 b = - \frac{384}{7}$

$a = - b + 48$

$c = 8$

$- 2 b = - \frac{384}{7}$

$a = - b + 48$

$c = 8$

Step 2.3.5.3

Divide each term in $- 2 b = - \frac{384}{7}$ by $- 2$ and simplify.

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Step 2.3.5.3.1

Divide each term in $- 2 b = - \frac{384}{7}$ by $- 2$ .

$\frac{- 2 b}{- 2} = \frac{- \frac{384}{7}}{- 2}$

$a = - b + 48$

$c = 8$

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.

$\frac{- 2 b}{- 2} = \frac{- \frac{384}{7}}{- 2}$

$a = - b + 48$

$c = 8$

Step 2.3.5.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{- \frac{384}{7}}{- 2}$

$a = - b + 48$

$c = 8$

$b = \frac{- \frac{384}{7}}{- 2}$

$a = - b + 48$

$c = 8$

$b = \frac{- \frac{384}{7}}{- 2}$

$a = - b + 48$

$c = 8$

Step 2.3.5.3.3

Simplify the right side.

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Step 2.3.5.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$b = - \frac{384}{7} \cdot \frac{1}{- 2}$

$a = - b + 48$

$c = 8$

Step 2.3.5.3.3.2

Cancel the common factor of $2$ .

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Step 2.3.5.3.3.2.1

Move the leading negative in $- \frac{384}{7}$ into the numerator.

$b = \frac{- 384}{7} \cdot \frac{1}{- 2}$

$a = - b + 48$

$c = 8$

Step 2.3.5.3.3.2.2

Factor $2$ out of $- 384$ .

$b = \frac{2 (- 192)}{7} \cdot \frac{1}{- 2}$

$a = - b + 48$

$c = 8$

Step 2.3.5.3.3.2.3

Factor $2$ out of $- 2$ .

$b = \frac{2 \cdot - 192}{7} \cdot \frac{1}{2 \cdot - 1}$

$a = - b + 48$

$c = 8$

Step 2.3.5.3.3.2.4

Cancel the common factor.

$b = \frac{2 \cdot - 192}{7} \cdot \frac{1}{2 \cdot - 1}$

$a = - b + 48$

$c = 8$

Step 2.3.5.3.3.2.5

Rewrite the expression.

$b = \frac{- 192}{7} \cdot \frac{1}{- 1}$

$a = - b + 48$

$c = 8$

$b = \frac{- 192}{7} \cdot \frac{1}{- 1}$

$a = - b + 48$

$c = 8$

Step 2.3.5.3.3.3

Multiply $\frac{- 192}{7}$ by $\frac{1}{- 1}$ .

$b = \frac{- 192}{7 \cdot - 1}$

$a = - b + 48$

$c = 8$

Step 2.3.5.3.3.4

Multiply $7$ by $- 1$ .

$b = \frac{- 192}{- 7}$

$a = - b + 48$

$c = 8$

Step 2.3.5.3.3.5

Dividing two negative values results in a positive value.

$b = \frac{192}{7}$

$a = - b + 48$

$c = 8$

$b = \frac{192}{7}$

$a = - b + 48$

$c = 8$

$b = \frac{192}{7}$

$a = - b + 48$

$c = 8$

$b = \frac{192}{7}$

$a = - b + 48$

$c = 8$

Step 2.3.6

Replace all occurrences of $b$ with $\frac{192}{7}$ in each equation.

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Step 2.3.6.1

Replace all occurrences of $b$ in $a = - b + 48$ with $\frac{192}{7}$ .

$a = - (\frac{192}{7}) + 48$

$b = \frac{192}{7}$

$c = 8$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $- (\frac{192}{7}) + 48$ .

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Step 2.3.6.2.1.1

To write $48$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$a = - \frac{192}{7} + 48 \cdot \frac{7}{7}$

$b = \frac{192}{7}$

$c = 8$

Step 2.3.6.2.1.2

Combine $48$ and $\frac{7}{7}$ .

$a = - \frac{192}{7} + \frac{48 \cdot 7}{7}$

$b = \frac{192}{7}$

$c = 8$

Step 2.3.6.2.1.3

Combine the numerators over the common denominator.

$a = \frac{- 192 + 48 \cdot 7}{7}$

$b = \frac{192}{7}$

$c = 8$

Step 2.3.6.2.1.4

Simplify the numerator.

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Step 2.3.6.2.1.4.1

Multiply $48$ by $7$ .

$a = \frac{- 192 + 336}{7}$

$b = \frac{192}{7}$

$c = 8$

Step 2.3.6.2.1.4.2

Add $- 192$ and $336$ .

$a = \frac{144}{7}$

$b = \frac{192}{7}$

$c = 8$

$a = \frac{144}{7}$

$b = \frac{192}{7}$

$c = 8$

$a = \frac{144}{7}$

$b = \frac{192}{7}$

$c = 8$

$a = \frac{144}{7}$

$b = \frac{192}{7}$

$c = 8$

$a = \frac{144}{7}$

$b = \frac{192}{7}$

$c = 8$

Step 2.3.7

List all of the solutions.

$a = \frac{144}{7}, b = \frac{192}{7}, c = 8$

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 $a = \frac{144}{7}$ , $b = \frac{192}{7}$ , $c = 8$ , 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 $2$ .

$y = \frac{144}{7} \cdot 1 + (\frac{192}{7}) \cdot (- 1) + 8$

Step 2.4.1.1.2

Multiply $\frac{144}{7}$ by $1$ .

$y = \frac{144}{7} + (\frac{192}{7}) \cdot (- 1) + 8$

Step 2.4.1.1.3

Multiply $(\frac{192}{7}) (- 1)$ .

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Step 2.4.1.1.3.1

Combine $\frac{192}{7}$ and $- 1$ .

$y = \frac{144}{7} + \frac{192 \cdot - 1}{7} + 8$

Step 2.4.1.1.3.2

Multiply $192$ by $- 1$ .

$y = \frac{144}{7} + \frac{- 192}{7} + 8$

Step 2.4.1.1.4

Move the negative in front of the fraction.

$y = \frac{144}{7} - \frac{192}{7} + 8$

Step 2.4.1.2

Combine fractions.

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Step 2.4.1.2.1

Combine the numerators over the common denominator.

$y = 8 + \frac{144 - 192}{7}$

Step 2.4.1.2.2

Simplify the expression.

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Step 2.4.1.2.2.1

Subtract $192$ from $144$ .

$y = 8 + \frac{- 48}{7}$

Step 2.4.1.2.2.2

Move the negative in front of the fraction.

$y = 8 - \frac{48}{7}$

Step 2.4.1.3

To write $8$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = 8 \cdot \frac{7}{7} - \frac{48}{7}$

Step 2.4.1.4

Combine $8$ and $\frac{7}{7}$ .

$y = \frac{8 \cdot 7}{7} - \frac{48}{7}$

Step 2.4.1.5

Combine the numerators over the common denominator.

$y = \frac{8 \cdot 7 - 48}{7}$

Step 2.4.1.6

Simplify the numerator.

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Step 2.4.1.6.1

Multiply $8$ by $7$ .

$y = \frac{56 - 48}{7}$

Step 2.4.1.6.2

Subtract $48$ from $56$ .

$y = \frac{8}{7}$

Step 2.4.2

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = - 1$ . This check passes since $y = \frac{8}{7}$ and $y = \frac{8}{7}$ .

$\frac{8}{7} = \frac{8}{7}$

Step 2.4.3

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = \frac{144}{7}$ , $b = \frac{192}{7}$ , $c = 8$ , and $x = 0$ .

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

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

$y = \frac{144}{7} \cdot 0 + (\frac{192}{7}) \cdot (0) + 8$

Step 2.4.3.1.2

Multiply $\frac{144}{7}$ by $0$ .

$y = 0 + (\frac{192}{7}) \cdot (0) + 8$

Step 2.4.3.1.3

Multiply $\frac{192}{7}$ by $0$ .

$y = 0 + 0 + 8$

Step 2.4.3.2

Simplify by adding numbers.

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Step 2.4.3.2.1

Add $0$ and $0$ .

$y = 0 + 8$

Step 2.4.3.2.2

Add $0$ and $8$ .

$y = 8$

Step 2.4.4

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 0$ . This check passes since $y = 8$ and $y = 8$ .

$8 = 8$

Step 2.4.5

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = \frac{144}{7}$ , $b = \frac{192}{7}$ , $c = 8$ , and $x = 1$ .

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Step 2.4.5.1

Simplify each term.

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Step 2.4.5.1.1

One to any power is one.

$y = \frac{144}{7} \cdot 1 + (\frac{192}{7}) \cdot (1) + 8$

Step 2.4.5.1.2

Multiply $\frac{144}{7}$ by $1$ .

$y = \frac{144}{7} + (\frac{192}{7}) \cdot (1) + 8$

Step 2.4.5.1.3

Multiply $\frac{192}{7}$ by $1$ .

$y = \frac{144}{7} + \frac{192}{7} + 8$

Step 2.4.5.2

Combine fractions.

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Step 2.4.5.2.1

Combine the numerators over the common denominator.

$y = 8 + \frac{144 + 192}{7}$

Step 2.4.5.2.2

Simplify the expression.

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Step 2.4.5.2.2.1

Add $144$ and $192$ .

$y = 8 + \frac{336}{7}$

Step 2.4.5.2.2.2

Divide $336$ by $7$ .

$y = 8 + 48$

Step 2.4.5.2.2.3

Add $8$ and $48$ .

$y = 56$

Step 2.4.6

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = 56$ and $y = 56$ .

$56 = 56$

Step 2.4.7

Since $y = y$ for the corresponding $x$ values, the function is quadratic.

The function is quadratic

Step 3

Since all $y = y$ , the function is quadratic and follows the form $y = \frac{144 x^{2}}{7} + \frac{192 x}{7} + 8$ .

$y = \frac{144 x^{2}}{7} + \frac{192 x}{7} + 8$