Precalculus Examples

$y = \frac{1}{2} x^{2} - x - 6$

Step 1

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

$y = \frac{x^{2}}{2} - x - 6$

Step 2

Find the properties of the given parabola.

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Step 2.1

Rewrite the equation in vertex form.

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Step 2.1.1

Complete the square for $\frac{x^{2}}{2} - x - 6$ .

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Step 2.1.1.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

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

$b = - 1$

$c = - 6$

Step 2.1.1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 2.1.1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 2.1.1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 1}{2 (\frac{1}{2})}$

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

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

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

Step 2.1.1.3.2.2

Divide $2$ by $2$ .

$d = \frac{- 1}{1}$

Step 2.1.1.3.2.3

Divide $- 1$ by $1$ .

$d = - 1$

Step 2.1.1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 2.1.1.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - 6 - \frac{{(- 1)}^{2}}{4 (\frac{1}{2})}$

Step 2.1.1.4.2

Simplify the right side.

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Step 2.1.1.4.2.1

Simplify each term.

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Step 2.1.1.4.2.1.1

Raise $- 1$ to the power of $2$ .

$e = - 6 - \frac{1}{4 (\frac{1}{2})}$

Step 2.1.1.4.2.1.2

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

$e = - 6 - \frac{1}{\frac{4}{2}}$

Step 2.1.1.4.2.1.3

Divide $4$ by $2$ .

$e = - 6 - \frac{1}{2}$

Step 2.1.1.4.2.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = - 6 \cdot \frac{2}{2} - \frac{1}{2}$

Step 2.1.1.4.2.3

Combine $- 6$ and $\frac{2}{2}$ .

$e = \frac{- 6 \cdot 2}{2} - \frac{1}{2}$

Step 2.1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{- 6 \cdot 2 - 1}{2}$

Step 2.1.1.4.2.5

Simplify the numerator.

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Step 2.1.1.4.2.5.1

Multiply $- 6$ by $2$ .

$e = \frac{- 12 - 1}{2}$

Step 2.1.1.4.2.5.2

Subtract $1$ from $- 12$ .

$e = \frac{- 13}{2}$

Step 2.1.1.4.2.6

Move the negative in front of the fraction.

$e = - \frac{13}{2}$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(x - 1)}^{2} - \frac{13}{2}$ .

$\frac{1}{2} {(x - 1)}^{2} - \frac{13}{2}$

Step 2.1.2

Set $y$ equal to the new right side.

$y = \frac{1}{2} \cdot {(x - 1)}^{2} - \frac{13}{2}$

Step 2.2

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

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

$h = 1$

$k = - \frac{13}{2}$

Step 2.3

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Step 2.4

Find the vertex $(h, k)$ .

$(1, - \frac{13}{2})$

Step 2.5

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

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Step 2.5.1

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

$\frac{1}{4 a}$

Step 2.5.2

Substitute the value of $a$ into the formula.

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

Step 2.5.3

Simplify.

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Step 2.5.3.1

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

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

Step 2.5.3.2

Divide $4$ by $2$ .

$\frac{1}{2}$

Step 2.6

Find the focus.

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Step 2.6.1

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 2.6.2

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

$(1, - 6)$

Step 2.7

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

$x = 1$

Step 2.8

Find the directrix.

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Step 2.8.1

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 2.8.2

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

$y = - 7$

Step 2.9

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

Direction: Opens Up

Vertex: $(1, - \frac{13}{2})$

Focus: $(1, - 6)$

Axis of Symmetry: $x = 1$

Directrix: $y = - 7$

Direction: Opens Up

Vertex: $(1, - \frac{13}{2})$

Focus: $(1, - 6)$

Axis of Symmetry: $x = 1$

Directrix: $y = - 7$

Step 3

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 3.1

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

$f (0) = \frac{{(0)}^{2}}{2} - (0) - 6$

Step 3.2

Simplify the result.

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Step 3.2.1

Find the common denominator.

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Step 3.2.1.1

Write $- (0)$ as a fraction with denominator $1$ .

$f (0) = \frac{{(0)}^{2}}{2} + \frac{- (0)}{1} - 6$

Step 3.2.1.2

Multiply $\frac{- (0)}{1}$ by $\frac{2}{2}$ .

$f (0) = \frac{{(0)}^{2}}{2} + \frac{- (0)}{1} \cdot \frac{2}{2} - 6$

Step 3.2.1.3

Multiply $\frac{- (0)}{1}$ by $\frac{2}{2}$ .

$f (0) = \frac{{(0)}^{2}}{2} + \frac{0 \cdot 2}{2} - 6$

Step 3.2.1.4

Write $- 6$ as a fraction with denominator $1$ .

$f (0) = \frac{{(0)}^{2}}{2} + \frac{0 \cdot 2}{2} + \frac{- 6}{1}$

Step 3.2.1.5

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$f (0) = \frac{{(0)}^{2}}{2} + \frac{0 \cdot 2}{2} + \frac{- 6}{1} \cdot \frac{2}{2}$

Step 3.2.1.6

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$f (0) = \frac{{(0)}^{2}}{2} + \frac{0 \cdot 2}{2} + \frac{- 6 \cdot 2}{2}$

Step 3.2.2

Combine the numerators over the common denominator.

$f (0) = \frac{{(0)}^{2} + 0 \cdot 2 - 6 \cdot 2}{2}$

Step 3.2.3

Simplify each term.

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Step 3.2.3.1

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

$f (0) = \frac{0 + 0 \cdot 2 - 6 \cdot 2}{2}$

Step 3.2.3.2

Multiply $- (0) \cdot 2$ .

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Step 3.2.3.2.1

Multiply $- 1$ by $0$ .

$f (0) = \frac{0 + 0 \cdot 2 - 6 \cdot 2}{2}$

Step 3.2.3.2.2

Multiply $0$ by $2$ .

$f (0) = \frac{0 + 0 - 6 \cdot 2}{2}$

Step 3.2.3.3

Multiply $- 6$ by $2$ .

$f (0) = \frac{0 + 0 - 12}{2}$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $0$ and $0$ .

$f (0) = \frac{0 - 12}{2}$

Step 3.2.4.2

Subtract $12$ from $0$ .

$f (0) = \frac{- 12}{2}$

Step 3.2.4.3

Divide $- 12$ by $2$ .

$f (0) = - 6$

Step 3.2.5

The final answer is $- 6$ .

$- 6$

Step 3.3

The $y$ value at $x = 0$ is $- 6$ .

$y = - 6$

Step 3.4

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

$f (- 1) = \frac{{(- 1)}^{2}}{2} - (- 1) - 6$

Step 3.5

Simplify the result.

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Step 3.5.1

Find the common denominator.

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Step 3.5.1.1

Write $- (- 1)$ as a fraction with denominator $1$ .

$f (- 1) = \frac{{(- 1)}^{2}}{2} + \frac{- (- 1)}{1} - 6$

Step 3.5.1.2

Multiply $\frac{- (- 1)}{1}$ by $\frac{2}{2}$ .

$f (- 1) = \frac{{(- 1)}^{2}}{2} + \frac{- (- 1)}{1} \cdot \frac{2}{2} - 6$

Step 3.5.1.3

Multiply $\frac{- (- 1)}{1}$ by $\frac{2}{2}$ .

$f (- 1) = \frac{{(- 1)}^{2}}{2} + \frac{1 \cdot 2}{2} - 6$

Step 3.5.1.4

Write $- 6$ as a fraction with denominator $1$ .

$f (- 1) = \frac{{(- 1)}^{2}}{2} + \frac{1 \cdot 2}{2} + \frac{- 6}{1}$

Step 3.5.1.5

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$f (- 1) = \frac{{(- 1)}^{2}}{2} + \frac{1 \cdot 2}{2} + \frac{- 6}{1} \cdot \frac{2}{2}$

Step 3.5.1.6

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$f (- 1) = \frac{{(- 1)}^{2}}{2} + \frac{1 \cdot 2}{2} + \frac{- 6 \cdot 2}{2}$

Step 3.5.2

Combine the numerators over the common denominator.

$f (- 1) = \frac{{(- 1)}^{2} + 1 \cdot 2 - 6 \cdot 2}{2}$

Step 3.5.3

Simplify each term.

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Step 3.5.3.1

Raise $- 1$ to the power of $2$ .

$f (- 1) = \frac{1 + 1 \cdot 2 - 6 \cdot 2}{2}$

Step 3.5.3.2

Multiply $- (- 1) \cdot 2$ .

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Step 3.5.3.2.1

Multiply $- 1$ by $- 1$ .

$f (- 1) = \frac{1 + 1 \cdot 2 - 6 \cdot 2}{2}$

Step 3.5.3.2.2

Multiply $2$ by $1$ .

$f (- 1) = \frac{1 + 2 - 6 \cdot 2}{2}$

Step 3.5.3.3

Multiply $- 6$ by $2$ .

$f (- 1) = \frac{1 + 2 - 12}{2}$

Step 3.5.4

Simplify the expression.

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Step 3.5.4.1

Add $1$ and $2$ .

$f (- 1) = \frac{3 - 12}{2}$

Step 3.5.4.2

Subtract $12$ from $3$ .

$f (- 1) = \frac{- 9}{2}$

Step 3.5.4.3

Move the negative in front of the fraction.

$f (- 1) = - \frac{9}{2}$

Step 3.5.5

The final answer is $- \frac{9}{2}$ .

$- \frac{9}{2}$

Step 3.6

The $y$ value at $x = - 1$ is $- \frac{9}{2}$ .

$y = - \frac{9}{2}$

Step 3.7

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

$f (2) = \frac{{(2)}^{2}}{2} - (2) - 6$

Step 3.8

Simplify the result.

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Step 3.8.1

Find the common denominator.

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Step 3.8.1.1

Write $- (2)$ as a fraction with denominator $1$ .

$f (2) = \frac{{(2)}^{2}}{2} + \frac{- (2)}{1} - 6$

Step 3.8.1.2

Multiply $\frac{- (2)}{1}$ by $\frac{2}{2}$ .

$f (2) = \frac{{(2)}^{2}}{2} + \frac{- (2)}{1} \cdot \frac{2}{2} - 6$

Step 3.8.1.3

Multiply $\frac{- (2)}{1}$ by $\frac{2}{2}$ .

$f (2) = \frac{{(2)}^{2}}{2} + \frac{- 2 \cdot 2}{2} - 6$

Step 3.8.1.4

Write $- 6$ as a fraction with denominator $1$ .

$f (2) = \frac{{(2)}^{2}}{2} + \frac{- 2 \cdot 2}{2} + \frac{- 6}{1}$

Step 3.8.1.5

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$f (2) = \frac{{(2)}^{2}}{2} + \frac{- 2 \cdot 2}{2} + \frac{- 6}{1} \cdot \frac{2}{2}$

Step 3.8.1.6

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$f (2) = \frac{{(2)}^{2}}{2} + \frac{- 2 \cdot 2}{2} + \frac{- 6 \cdot 2}{2}$

Step 3.8.2

Combine the numerators over the common denominator.

$f (2) = \frac{{(2)}^{2} - 2 \cdot 2 - 6 \cdot 2}{2}$

Step 3.8.3

Simplify each term.

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Step 3.8.3.1

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

$f (2) = \frac{4 - 2 \cdot 2 - 6 \cdot 2}{2}$

Step 3.8.3.2

Multiply $- (2) \cdot 2$ .

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Step 3.8.3.2.1

Multiply $- 1$ by $2$ .

$f (2) = \frac{4 - 2 \cdot 2 - 6 \cdot 2}{2}$

Step 3.8.3.2.2

Multiply $- 2$ by $2$ .

$f (2) = \frac{4 - 4 - 6 \cdot 2}{2}$

Step 3.8.3.3

Multiply $- 6$ by $2$ .

$f (2) = \frac{4 - 4 - 12}{2}$

Step 3.8.4

Simplify the expression.

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Step 3.8.4.1

Subtract $4$ from $4$ .

$f (2) = \frac{0 - 12}{2}$

Step 3.8.4.2

Subtract $12$ from $0$ .

$f (2) = \frac{- 12}{2}$

Step 3.8.4.3

Divide $- 12$ by $2$ .

$f (2) = - 6$

Step 3.8.5

The final answer is $- 6$ .

$- 6$

Step 3.9

The $y$ value at $x = 2$ is $- 6$ .

$y = - 6$

Step 3.10

Replace the variable $x$ with $4$ in the expression.

$f (4) = \frac{{(4)}^{2}}{2} - (4) - 6$

Step 3.11

Simplify the result.

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Step 3.11.1

Find the common denominator.

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Step 3.11.1.1

Write $- (4)$ as a fraction with denominator $1$ .

$f (4) = \frac{{(4)}^{2}}{2} + \frac{- (4)}{1} - 6$

Step 3.11.1.2

Multiply $\frac{- (4)}{1}$ by $\frac{2}{2}$ .

$f (4) = \frac{{(4)}^{2}}{2} + \frac{- (4)}{1} \cdot \frac{2}{2} - 6$

Step 3.11.1.3

Multiply $\frac{- (4)}{1}$ by $\frac{2}{2}$ .

$f (4) = \frac{{(4)}^{2}}{2} + \frac{- 4 \cdot 2}{2} - 6$

Step 3.11.1.4

Write $- 6$ as a fraction with denominator $1$ .

$f (4) = \frac{{(4)}^{2}}{2} + \frac{- 4 \cdot 2}{2} + \frac{- 6}{1}$

Step 3.11.1.5

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$f (4) = \frac{{(4)}^{2}}{2} + \frac{- 4 \cdot 2}{2} + \frac{- 6}{1} \cdot \frac{2}{2}$

Step 3.11.1.6

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$f (4) = \frac{{(4)}^{2}}{2} + \frac{- 4 \cdot 2}{2} + \frac{- 6 \cdot 2}{2}$

Step 3.11.2

Combine the numerators over the common denominator.

$f (4) = \frac{{(4)}^{2} - 4 \cdot 2 - 6 \cdot 2}{2}$

Step 3.11.3

Simplify each term.

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Step 3.11.3.1

Raise $4$ to the power of $2$ .

$f (4) = \frac{16 - 4 \cdot 2 - 6 \cdot 2}{2}$

Step 3.11.3.2

Multiply $- (4) \cdot 2$ .

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Step 3.11.3.2.1

Multiply $- 1$ by $4$ .

$f (4) = \frac{16 - 4 \cdot 2 - 6 \cdot 2}{2}$

Step 3.11.3.2.2

Multiply $- 4$ by $2$ .

$f (4) = \frac{16 - 8 - 6 \cdot 2}{2}$

Step 3.11.3.3

Multiply $- 6$ by $2$ .

$f (4) = \frac{16 - 8 - 12}{2}$

Step 3.11.4

Simplify the expression.

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Step 3.11.4.1

Subtract $8$ from $16$ .

$f (4) = \frac{8 - 12}{2}$

Step 3.11.4.2

Subtract $12$ from $8$ .

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

Step 3.11.4.3

Divide $- 4$ by $2$ .

$f (4) = - 2$

Step 3.11.5

The final answer is $- 2$ .

$- 2$

Step 3.12

The $y$ value at $x = 4$ is $- 2$ .

$y = - 2$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - \frac{9}{2} \\ 0 & - 6 \\ 1 & - \frac{13}{2} \\ 2 & - 6 \\ 4 & - 2 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(1, - \frac{13}{2})$

Focus: $(1, - 6)$

Axis of Symmetry: $x = 1$

Directrix: $y = - 7$

$\begin{array}{c} x & y \\ - 1 & - \frac{9}{2} \\ 0 & - 6 \\ 1 & - \frac{13}{2} \\ 2 & - 6 \\ 4 & - 2 \end{array}$

Step 5