Pre-Algebra Examples

$f (x) = - \frac{1}{2} \cdot {(x + 1)}^{2} - 3$

Step 1

Find the properties of the given parabola.

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Step 1.1

Isolate $y$ to the left side of the equation.

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Step 1.1.1

Combine ${(x + 1)}^{2}$ and $\frac{1}{2}$ .

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

Step 1.1.2

Reorder terms.

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

Step 1.2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

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

$h = - 1$

$k = - 3$

Step 1.3

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 1.4

Find the vertex $(h, k)$ .

$(- 1, - 3)$

Step 1.5

Find $p$ , the distance from the vertex to the focus.

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Step 1.5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot (- \frac{1}{2})}$

Step 1.5.3

Simplify.

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Step 1.5.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 1.5.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{2})}$

Step 1.5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{2})}$

Step 1.5.3.2

Combine $4$ and $\frac{1}{2}$ .

$- \frac{1}{\frac{4}{2}}$

Step 1.5.3.3

Divide $4$ by $2$ .

$- \frac{1}{2}$

Step 1.6

Find the focus.

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Step 1.6.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 1.6.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(- 1, - \frac{7}{2})$

Step 1.7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = - 1$

Step 1.8

Find the directrix.

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Step 1.8.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 1.8.2

Substitute the known values of $p$ and $k$ into the formula and simplify.

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

Step 1.9

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Down

Vertex: $(- 1, - 3)$

Focus: $(- 1, - \frac{7}{2})$

Axis of Symmetry: $x = - 1$

Directrix: $y = - \frac{5}{2}$

Direction: Opens Down

Vertex: $(- 1, - 3)$

Focus: $(- 1, - \frac{7}{2})$

Axis of Symmetry: $x = - 1$

Directrix: $y = - \frac{5}{2}$

Step 2

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

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Step 2.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = - \frac{{(- 3)}^{2}}{2} - (- 3) - \frac{7}{2}$

Step 2.2

Simplify the result.

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Step 2.2.1

Combine the numerators over the common denominator.

$f (- 3) = - (- 3) + \frac{- {(- 3)}^{2} - 7}{2}$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

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

$f (- 3) = - (- 3) + \frac{- 1 \cdot 9 - 7}{2}$

Step 2.2.2.2

Multiply $- 1$ by $9$ .

$f (- 3) = - (- 3) + \frac{- 9 - 7}{2}$

Step 2.2.3

Subtract $7$ from $- 9$ .

$f (- 3) = - (- 3) + \frac{- 16}{2}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Multiply $- 1$ by $- 3$ .

$f (- 3) = 3 + \frac{- 16}{2}$

Step 2.2.4.2

Divide $- 16$ by $2$ .

$f (- 3) = 3 - 8$

Step 2.2.5

Subtract $8$ from $3$ .

$f (- 3) = - 5$

Step 2.2.6

The final answer is $- 5$ .

$- 5$

Step 2.3

The $y$ value at $x = - 3$ is $- 5$ .

$y = - 5$

Step 2.4

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = - \frac{{(- 2)}^{2}}{2} - (- 2) - \frac{7}{2}$

Step 2.5

Simplify the result.

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Step 2.5.1

Combine the numerators over the common denominator.

$f (- 2) = - (- 2) + \frac{- {(- 2)}^{2} - 7}{2}$

Step 2.5.2

Simplify each term.

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Step 2.5.2.1

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

$f (- 2) = - (- 2) + \frac{- 1 \cdot 4 - 7}{2}$

Step 2.5.2.2

Multiply $- 1$ by $4$ .

$f (- 2) = - (- 2) + \frac{- 4 - 7}{2}$

Step 2.5.3

Subtract $7$ from $- 4$ .

$f (- 2) = - (- 2) + \frac{- 11}{2}$

Step 2.5.4

Simplify each term.

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Step 2.5.4.1

Multiply $- 1$ by $- 2$ .

$f (- 2) = 2 + \frac{- 11}{2}$

Step 2.5.4.2

Move the negative in front of the fraction.

$f (- 2) = 2 - \frac{11}{2}$

Step 2.5.5

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

$f (- 2) = 2 \cdot \frac{2}{2} - \frac{11}{2}$

Step 2.5.6

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

$f (- 2) = \frac{2 \cdot 2}{2} - \frac{11}{2}$

Step 2.5.7

Combine the numerators over the common denominator.

$f (- 2) = \frac{2 \cdot 2 - 11}{2}$

Step 2.5.8

Simplify the numerator.

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Step 2.5.8.1

Multiply $2$ by $2$ .

$f (- 2) = \frac{4 - 11}{2}$

Step 2.5.8.2

Subtract $11$ from $4$ .

$f (- 2) = \frac{- 7}{2}$

Step 2.5.9

Move the negative in front of the fraction.

$f (- 2) = - \frac{7}{2}$

Step 2.5.10

The final answer is $- \frac{7}{2}$ .

$- \frac{7}{2}$

Step 2.6

The $y$ value at $x = - 2$ is $- \frac{7}{2}$ .

$y = - \frac{7}{2}$

Step 2.7

Replace the variable $x$ with $1$ in the expression.

$f (1) = - \frac{{(1)}^{2}}{2} - (1) - \frac{7}{2}$

Step 2.8

Simplify the result.

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Step 2.8.1

Combine the numerators over the common denominator.

$f (1) = - (1) + \frac{- {(1)}^{2} - 7}{2}$

Step 2.8.2

Simplify each term.

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Step 2.8.2.1

One to any power is one.

$f (1) = - (1) + \frac{- 1 \cdot 1 - 7}{2}$

Step 2.8.2.2

Multiply $- 1$ by $1$ .

$f (1) = - (1) + \frac{- 1 - 7}{2}$

Step 2.8.3

Subtract $7$ from $- 1$ .

$f (1) = - (1) + \frac{- 8}{2}$

Step 2.8.4

Simplify each term.

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Step 2.8.4.1

Multiply $- 1$ by $1$ .

$f (1) = - 1 + \frac{- 8}{2}$

Step 2.8.4.2

Divide $- 8$ by $2$ .

$f (1) = - 1 - 4$

Step 2.8.5

Subtract $4$ from $- 1$ .

$f (1) = - 5$

Step 2.8.6

The final answer is $- 5$ .

$- 5$

Step 2.9

The $y$ value at $x = 1$ is $- 5$ .

$y = - 5$

Step 2.10

Replace the variable $x$ with $0$ in the expression.

$f (0) = - \frac{{(0)}^{2}}{2} - (0) - \frac{7}{2}$

Step 2.11

Simplify the result.

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Step 2.11.1

Combine the numerators over the common denominator.

$f (0) = - (0) + \frac{- {(0)}^{2} - 7}{2}$

Step 2.11.2

Simplify each term.

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Step 2.11.2.1

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

$f (0) = - (0) + \frac{- 0 - 7}{2}$

Step 2.11.2.2

Multiply $- 1$ by $0$ .

$f (0) = - (0) + \frac{0 - 7}{2}$

Step 2.11.3

Subtract $7$ from $0$ .

$f (0) = - (0) + \frac{- 7}{2}$

Step 2.11.4

Simplify each term.

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Step 2.11.4.1

Multiply $- 1$ by $0$ .

$f (0) = 0 + \frac{- 7}{2}$

Step 2.11.4.2

Move the negative in front of the fraction.

$f (0) = 0 - \frac{7}{2}$

Step 2.11.5

Subtract $\frac{7}{2}$ from $0$ .

$f (0) = - \frac{7}{2}$

Step 2.11.6

The final answer is $- \frac{7}{2}$ .

$- \frac{7}{2}$

Step 2.12

The $y$ value at $x = 0$ is $- \frac{7}{2}$ .

$y = - \frac{7}{2}$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & - 5 \\ - 2 & - \frac{7}{2} \\ - 1 & - 3 \\ 0 & - \frac{7}{2} \\ 1 & - 5 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- 1, - 3)$

Focus: $(- 1, - \frac{7}{2})$

Axis of Symmetry: $x = - 1$

Directrix: $y = - \frac{5}{2}$

$\begin{array}{c} x & y \\ - 3 & - 5 \\ - 2 & - \frac{7}{2} \\ - 1 & - 3 \\ 0 & - \frac{7}{2} \\ 1 & - 5 \end{array}$

Step 4