Algebra Examples

\begin{matrix} x & y 0 & 2 1 & 6 2 & 18 \end{matrix}

$\begin{array}{c} x & y \\ 0 & 2 \\ 1 & 6 \\ 2 & 18 \end{array}$

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

$y = a x + b$ .

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

$y = a x + b$ .

$2 = a (0) + b 6 = a (1) + b 18 = a (2) + b$

Step 1.3

Calculate the values of

$a$ and

$b$ .

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Step 1.3.1

Rewrite the equation as

$b = 2$ .

$b = 2$

$6 = a + b$

$18 = a (2) + b$

Step 1.3.2

Replace all occurrences of

$b$ with

$2$ in each equation.

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Step 1.3.2.1

Replace all occurrences of

$b$ in

$6 = a + b$ with

$2$ .

$6 = a + 2$

$b = 2$

$18 = a (2) + b$

Step 1.3.2.2

Simplify the left side.

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Step 1.3.2.2.1

Remove parentheses.

$6 = a + 2$

$b = 2$

$18 = a (2) + b$

$6 = a + 2$

$b = 2$

$18 = a (2) + b$

Step 1.3.2.3

Replace all occurrences of

$b$ in

$18 = a (2) + b$ with

$2$ .

$18 = a (2) + 2$

$6 = a + 2$

$b = 2$

Step 1.3.2.4

Simplify

$18 = a (2) + 2$ .

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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.

$18 = a (2) + 2$

$6 = a + 2$

$b = 2$

$18 = a (2) + 2$

$6 = a + 2$

$b = 2$

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

$a$ .

$18 = 2 a + 2$

$6 = a + 2$

$b = 2$

$18 = 2 a + 2$

$6 = a + 2$

$b = 2$

$18 = 2 a + 2$

$6 = a + 2$

$b = 2$

$18 = 2 a + 2$

$6 = a + 2$

$b = 2$

Step 1.3.3

Solve for

$a$ in

$18 = 2 a + 2$ .

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Step 1.3.3.1

Rewrite the equation as

$2 a + 2 = 18$ .

$2 a + 2 = 18$

$6 = a + 2$

$b = 2$

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

$2$ from both sides of the equation.

$2 a = 18 - 2$

$6 = a + 2$

$b = 2$

Step 1.3.3.2.2

Subtract

$2$ from

$18$ .

$2 a = 16$

$6 = a + 2$

$b = 2$

$2 a = 16$

$6 = a + 2$

$b = 2$

Step 1.3.3.3

Divide each term in

$2 a = 16$ by

$2$ and simplify.

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Step 1.3.3.3.1

Divide each term in

$2 a = 16$ by

$2$ .

$\frac{2 a}{2} = \frac{16}{2}$

$6 = a + 2$

$b = 2$

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.

$\frac{2 a}{2} = \frac{16}{2}$

$6 = a + 2$

$b = 2$

Step 1.3.3.3.2.1.2

Divide

$a$ by

$1$ .

$a = \frac{16}{2}$

$6 = a + 2$

$b = 2$

$a = \frac{16}{2}$

$6 = a + 2$

$b = 2$

$a = \frac{16}{2}$

$6 = a + 2$

$b = 2$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide

$16$ by

$2$ .

$a = 8$

$6 = a + 2$

$b = 2$

$a = 8$

$6 = a + 2$

$b = 2$

$a = 8$

$6 = a + 2$

$b = 2$

$a = 8$

$6 = a + 2$

$b = 2$

Step 1.3.4

Replace all occurrences of

$a$ with

$8$ in each equation.

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Step 1.3.4.1

Replace all occurrences of

$a$ in

$6 = a + 2$ with

$8$ .

$6 = (8) + 2$

$a = 8$

$b = 2$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Add

$8$ and

$2$ .

$6 = 10$

$a = 8$

$b = 2$

$6 = 10$

$a = 8$

$b = 2$

$6 = 10$

$a = 8$

$b = 2$

Step 1.3.5

Since

$6 = 10$ 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

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

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Step 2.3.1.1

Rewrite the equation as

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

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

$6 = a + b + c$

$18 = a \cdot 2^{2} + b (2) + c$

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

$0$ .

$a \cdot 0 + c = 2$

$6 = a + b + c$

$18 = a \cdot 2^{2} + b (2) + c$

Step 2.3.1.2.1.2

Multiply

$a$ by

$0$ .

$0 + c = 2$

$6 = a + b + c$

$18 = a \cdot 2^{2} + b (2) + c$

$0 + c = 2$

$6 = a + b + c$

$18 = a \cdot 2^{2} + b (2) + c$

Step 2.3.1.2.2

Add

$0$ and

$c$ .

$c = 2$

$6 = a + b + c$

$18 = a \cdot 2^{2} + b (2) + c$

$c = 2$

$6 = a + b + c$

$18 = a \cdot 2^{2} + b (2) + c$

$c = 2$

$6 = a + b + c$

$18 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2

Replace all occurrences of

$c$ with

$2$ in each equation.

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Step 2.3.2.1

Replace all occurrences of

$c$ in

$6 = a + b + c$ with

$2$ .

$6 = a + b + 2$

$c = 2$

$18 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Remove parentheses.

$6 = a + b + 2$

$c = 2$

$18 = a \cdot 2^{2} + b (2) + c$

$6 = a + b + 2$

$c = 2$

$18 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2.3

Replace all occurrences of

$c$ in

$18 = a \cdot 2^{2} + b (2) + c$ with

$2$ .

$18 = a \cdot 2^{2} + b (2) + 2$

$6 = a + b + 2$

$c = 2$

Step 2.3.2.4

Simplify

$18 = a \cdot 2^{2} + b (2) + 2$ .

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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.

$18 = a \cdot 2^{2} + b (2) + 2$

$6 = a + b + 2$

$c = 2$

$18 = a \cdot 2^{2} + b (2) + 2$

$6 = a + b + 2$

$c = 2$

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

$2$ .

$18 = a \cdot 4 + b (2) + 2$

$6 = a + b + 2$

$c = 2$

Step 2.3.2.4.2.1.2

Move

$4$ to the left of

$a$ .

$18 = 4 \cdot a + b (2) + 2$

$6 = a + b + 2$

$c = 2$

Step 2.3.2.4.2.1.3

Move

$2$ to the left of

$b$ .

$18 = 4 a + 2 b + 2$

$6 = a + b + 2$

$c = 2$

$18 = 4 a + 2 b + 2$

$6 = a + b + 2$

$c = 2$

$18 = 4 a + 2 b + 2$

$6 = a + b + 2$

$c = 2$

$18 = 4 a + 2 b + 2$

$6 = a + b + 2$

$c = 2$

$18 = 4 a + 2 b + 2$

$6 = a + b + 2$

$c = 2$

Step 2.3.3

Solve for

$a$ in

$6 = a + b + 2$ .

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Step 2.3.3.1

Rewrite the equation as

$a + b + 2 = 6$ .

$a + b + 2 = 6$

$18 = 4 a + 2 b + 2$

$c = 2$

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.

$a + 2 = 6 - b$

$18 = 4 a + 2 b + 2$

$c = 2$

Step 2.3.3.2.2

Subtract

$2$ from both sides of the equation.

$a = 6 - b - 2$

$18 = 4 a + 2 b + 2$

$c = 2$

Step 2.3.3.2.3

Subtract

$2$ from

$6$ .

$a = - b + 4$

$18 = 4 a + 2 b + 2$

$c = 2$

$a = - b + 4$

$18 = 4 a + 2 b + 2$

$c = 2$

$a = - b + 4$

$18 = 4 a + 2 b + 2$

$c = 2$

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

$18 = 4 a + 2 b + 2$ with

$- b + 4$ .

$18 = 4 (- b + 4) + 2 b + 2$

$a = - b + 4$

$c = 2$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify

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

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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.

$18 = 4 (- b) + 4 \cdot 4 + 2 b + 2$

$a = - b + 4$

$c = 2$

Step 2.3.4.2.1.1.2

Multiply

$- 1$ by

$4$ .

$18 = - 4 b + 4 \cdot 4 + 2 b + 2$

$a = - b + 4$

$c = 2$

Step 2.3.4.2.1.1.3

Multiply

$4$ by

$4$ .

$18 = - 4 b + 16 + 2 b + 2$

$a = - b + 4$

$c = 2$

$18 = - 4 b + 16 + 2 b + 2$

$a = - b + 4$

$c = 2$

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

$2 b$ .

$18 = - 2 b + 16 + 2$

$a = - b + 4$

$c = 2$

Step 2.3.4.2.1.2.2

Add

$16$ and

$2$ .

$18 = - 2 b + 18$

$a = - b + 4$

$c = 2$

$18 = - 2 b + 18$

$a = - b + 4$

$c = 2$

$18 = - 2 b + 18$

$a = - b + 4$

$c = 2$

$18 = - 2 b + 18$

$a = - b + 4$

$c = 2$

$18 = - 2 b + 18$

$a = - b + 4$

$c = 2$

Step 2.3.5

Solve for

$b$ in

$18 = - 2 b + 18$ .

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Step 2.3.5.1

Rewrite the equation as

$- 2 b + 18 = 18$ .

$- 2 b + 18 = 18$

$a = - b + 4$

$c = 2$

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

$18$ from both sides of the equation.

$- 2 b = 18 - 18$

$a = - b + 4$

$c = 2$

Step 2.3.5.2.2

Subtract

$18$ from

$18$ .

$- 2 b = 0$

$a = - b + 4$

$c = 2$

$- 2 b = 0$

$a = - b + 4$

$c = 2$

Step 2.3.5.3

Divide each term in

$- 2 b = 0$ by

$- 2$ and simplify.

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Step 2.3.5.3.1

Divide each term in

$- 2 b = 0$ by

$- 2$ .

$\frac{- 2 b}{- 2} = \frac{0}{- 2}$

$a = - b + 4$

$c = 2$

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.

$\frac{- 2 b}{- 2} = \frac{0}{- 2}$

$a = - b + 4$

$c = 2$

Step 2.3.5.3.2.1.2

Divide

$b$ by

$1$ .

$b = \frac{0}{- 2}$

$a = - b + 4$

$c = 2$

$b = \frac{0}{- 2}$

$a = - b + 4$

$c = 2$

$b = \frac{0}{- 2}$

$a = - b + 4$

$c = 2$

Step 2.3.5.3.3

Simplify the right side.

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Step 2.3.5.3.3.1

Divide

$0$ by

$- 2$ .

$b = 0$

$a = - b + 4$

$c = 2$

$b = 0$

$a = - b + 4$

$c = 2$

$b = 0$

$a = - b + 4$

$c = 2$

$b = 0$

$a = - b + 4$

$c = 2$

Step 2.3.6

Replace all occurrences of

$b$ with

$0$ in each equation.

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Step 2.3.6.1

Replace all occurrences of

$b$ in

$a = - b + 4$ with

$0$ .

$a = - (0) + 4$

$b = 0$

$c = 2$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify

$- (0) + 4$ .

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Step 2.3.6.2.1.1

Multiply

$- 1$ by

$0$ .

$a = 0 + 4$

$b = 0$

$c = 2$

Step 2.3.6.2.1.2

Add

$0$ and

$4$ .

$a = 4$

$b = 0$

$c = 2$

$a = 4$

$b = 0$

$c = 2$

$a = 4$

$b = 0$

$c = 2$

$a = 4$

$b = 0$

$c = 2$

Step 2.3.7

List all of the solutions.

$a = 4, b = 0, c = 2$

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

$a = 4$ ,

$b = 0$ ,

$c = 2$ , 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

Raising

$0$ to any positive power yields

$0$ .

$y = 4 \cdot 0 + (0) \cdot (0) + 2$

Step 2.4.1.1.2

Multiply

$4$ by

$0$ .

$y = 0 + (0) \cdot (0) + 2$

Step 2.4.1.1.3

Multiply

$0$ by

$0$ .

$y = 0 + 0 + 2$

Step 2.4.1.2

Simplify by adding numbers.

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Step 2.4.1.2.1

Add

$0$ and

$0$ .

$y = 0 + 2$

Step 2.4.1.2.2

Add

$0$ and

$2$ .

$y = 2$

Step 2.4.2

If the table has a quadratic function rule,

$y = y$ for the corresponding

$x$ value,

$x = 0$ . This check passes since

$y = 2$ and

$y = 2$ .

$2 = 2$

Step 2.4.3

Calculate the value of

$y$ such that

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

$a = 4$ ,

$b = 0$ ,

$c = 2$ , and

$x = 1$ .

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

One to any power is one.

$y = 4 \cdot 1 + (0) \cdot (1) + 2$

Step 2.4.3.1.2

Multiply

$4$ by

$1$ .

$y = 4 + (0) \cdot (1) + 2$

Step 2.4.3.1.3

Multiply

$0$ by

$1$ .

$y = 4 + 0 + 2$

Step 2.4.3.2

Simplify by adding numbers.

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Step 2.4.3.2.1

Add

$4$ and

$0$ .

$y = 4 + 2$

Step 2.4.3.2.2

Add

$4$ and

$2$ .

$y = 6$

Step 2.4.4

If the table has a quadratic function rule,

$y = y$ for the corresponding

$x$ value,

$x = 1$ . This check passes since

$y = 6$ and

$y = 6$ .

$6 = 6$

Step 2.4.5

Calculate the value of

$y$ such that

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

$a = 4$ ,

$b = 0$ ,

$c = 2$ , and

$x = 2$ .

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Step 2.4.5.1

Simplify each term.

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Step 2.4.5.1.1

Raise

$2$ to the power of

$2$ .

$y = 4 \cdot 4 + (0) \cdot (2) + 2$

Step 2.4.5.1.2

Multiply

$4$ by

$4$ .

$y = 16 + (0) \cdot (2) + 2$

Step 2.4.5.1.3

Multiply

$0$ by

$2$ .

$y = 16 + 0 + 2$

Step 2.4.5.2

Simplify by adding numbers.

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Step 2.4.5.2.1

Add

$16$ and

$0$ .

$y = 16 + 2$

Step 2.4.5.2.2

Add

$16$ and

$2$ .

$y = 18$

Step 2.4.6

If the table has a quadratic function rule,

$y = y$ for the corresponding

$x$ value,

$x = 2$ . This check passes since

$y = 18$ and

$y = 18$ .

$18 = 18$

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

$y = 4 x^{2} + 2$ .

$y = 4 x^{2} + 2$