Algebra Examples

f (x) = - \frac{1}{2} x^{2}

$f (x) = - \frac{1}{2} x^{2}$

Step 1

Substitute

- 2

$- 2$ for

x

$x$ and find the result for

y

$y$ .

y = - \frac{1}{2} \cdot {(- 2)}^{2}

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

Step 2

Simplify

- \frac{1}{2} \cdot {(- 2)}^{2}

$- \frac{1}{2} \cdot {(- 2)}^{2}$ .

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Step 2.1

Raise

- 2

$- 2$ to the power of

2

$2$ .

y = - \frac{1}{2} \cdot 4

$y = - \frac{1}{2} \cdot 4$

Step 2.2

Cancel the common factor of

2

$2$ .

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Step 2.2.1

Move the leading negative in

- \frac{1}{2}

$- \frac{1}{2}$ into the numerator.

y = \frac{- 1}{2} \cdot 4

$y = \frac{- 1}{2} \cdot 4$

Step 2.2.2

Factor

2

$2$ out of

4

$4$ .

y = \frac{- 1}{2} \cdot (2 (2))

$y = \frac{- 1}{2} \cdot (2 (2))$

Step 2.2.3

Cancel the common factor.

$y = \frac{- 1}{2} \cdot (2 \cdot 2)$

Step 2.2.4

Rewrite the expression.

$y = - 1 \cdot 2$

Step 2.3

Multiply

$- 1$ by

$2$ .

$y = - 2$

Step 3

Substitute

$- 1$ for

$x$ and find the result for

$y$ .

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

Step 4

Simplify

$- \frac{1}{2} \cdot {(- 1)}^{2}$ .

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Step 4.1

Multiply

$- 1$ by

${(- 1)}^{2}$ by adding the exponents.

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Step 4.1.1

Move

${(- 1)}^{2}$ .

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

Step 4.1.2

Multiply

${(- 1)}^{2}$ by

$- 1$ .

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Step 4.1.2.1

Raise

$- 1$ to the power of

$1$ .

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

Step 4.1.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.1.3

Add

$2$ and

$1$ .

$y = {(- 1)}^{3} \frac{1}{2}$

Step 4.2

Raise

$- 1$ to the power of

$3$ .

$y = - \frac{1}{2}$

Step 5

Substitute

$0$ for

$x$ and find the result for

$y$ .

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

Step 6

Solve the equation for

$y$ .

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Step 6.1

Remove parentheses.

$y = - \frac{1}{2} \cdot 0^{2}$

Step 6.2

Remove parentheses.

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

Step 6.3

Simplify

$- \frac{1}{2} \cdot {(0)}^{2}$ .

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Step 6.3.1

Raising

$0$ to any positive power yields

$0$ .

$y = - \frac{1}{2} \cdot 0$

Step 6.3.2

Multiply

$- \frac{1}{2} \cdot 0$ .

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Step 6.3.2.1

Multiply

$0$ by

$- 1$ .

$y = 0 (\frac{1}{2})$

Step 6.3.2.2

Multiply

$0$ by

$\frac{1}{2}$ .

$y = 0$

Step 7

Substitute

$1$ for

$x$ and find the result for

$y$ .

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

Step 8

Solve the equation for

$y$ .

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Step 8.1

Remove parentheses.

$y = - \frac{1}{2} \cdot 1^{2}$

Step 8.2

Remove parentheses.

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

Step 8.3

Simplify

$- \frac{1}{2} \cdot {(1)}^{2}$ .

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Step 8.3.1

One to any power is one.

$y = - \frac{1}{2} \cdot 1$

Step 8.3.2

Multiply

$- 1$ by

$1$ .

$y = - \frac{1}{2}$

Step 9

Substitute

$2$ for

$x$ and find the result for

$y$ .

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

Step 10

Solve the equation for

$y$ .

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Step 10.1

Remove parentheses.

$y = - \frac{1}{2} \cdot 2^{2}$

Step 10.2

Remove parentheses.

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

Step 10.3

Simplify

$- \frac{1}{2} \cdot {(2)}^{2}$ .

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Step 10.3.1

Cancel the common factor of

$2$ .

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Step 10.3.1.1

Move the leading negative in

$- \frac{1}{2}$ into the numerator.

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

Step 10.3.1.2

Factor

$2$ out of

${(2)}^{2}$ .

$y = \frac{- 1}{2} \cdot (2 \cdot 2)$

Step 10.3.1.3

Cancel the common factor.

$y = \frac{- 1}{2} \cdot (2 \cdot 2)$

Step 10.3.1.4

Rewrite the expression.

$y = - 1 \cdot 2$

Step 10.3.2

Multiply

$- 1$ by

$2$ .

$y = - 2$

Step 11

This is a table of possible values to use when graphing the equation.

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

Step 12