Precalculus Examples

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

Step 1

Find the properties of the given parabola.

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Step 1.1

Subtract $2$ from both sides of the equation.

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

Step 1.2

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

$a = - 2$

$h = - 3$

$k = - 2$

Step 1.3

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

Opens Down

Step 1.4

Find the vertex $(h, k)$ .

$(- 3, - 2)$

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.

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

Step 1.5.3

Simplify.

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Step 1.5.3.1

Multiply $4$ by $- 2$ .

$\frac{1}{- 8}$

Step 1.5.3.2

Move the negative in front of the fraction.

$- \frac{1}{8}$

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.

$(- 3, - \frac{17}{8})$

Step 1.7

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

$x = - 3$

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.

$y = - \frac{15}{8}$

Step 1.9

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

Direction: Opens Down

Vertex: $(- 3, - 2)$

Focus: $(- 3, - \frac{17}{8})$

Axis of Symmetry: $x = - 3$

Directrix: $y = - \frac{15}{8}$

Direction: Opens Down

Vertex: $(- 3, - 2)$

Focus: $(- 3, - \frac{17}{8})$

Axis of Symmetry: $x = - 3$

Directrix: $y = - \frac{15}{8}$

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 $- 4$ in the expression.

$f (- 4) = - 2 {(- 4)}^{2} - 12 \cdot - 4 - 20$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

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

$f (- 4) = - 2 \cdot 16 - 12 \cdot - 4 - 20$

Step 2.2.1.2

Multiply $- 2$ by $16$ .

$f (- 4) = - 32 - 12 \cdot - 4 - 20$

Step 2.2.1.3

Multiply $- 12$ by $- 4$ .

$f (- 4) = - 32 + 48 - 20$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $- 32$ and $48$ .

$f (- 4) = 16 - 20$

Step 2.2.2.2

Subtract $20$ from $16$ .

$f (- 4) = - 4$

Step 2.2.3

The final answer is $- 4$ .

$- 4$

Step 2.3

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

$y = - 4$

Step 2.4

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = - 2 {(- 5)}^{2} - 12 \cdot - 5 - 20$

Step 2.5

Simplify the result.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Raise $- 5$ to the power of $2$ .

$f (- 5) = - 2 \cdot 25 - 12 \cdot - 5 - 20$

Step 2.5.1.2

Multiply $- 2$ by $25$ .

$f (- 5) = - 50 - 12 \cdot - 5 - 20$

Step 2.5.1.3

Multiply $- 12$ by $- 5$ .

$f (- 5) = - 50 + 60 - 20$

Step 2.5.2

Simplify by adding and subtracting.

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Step 2.5.2.1

Add $- 50$ and $60$ .

$f (- 5) = 10 - 20$

Step 2.5.2.2

Subtract $20$ from $10$ .

$f (- 5) = - 10$

Step 2.5.3

The final answer is $- 10$ .

$- 10$

Step 2.6

The $y$ value at $x = - 5$ is $- 10$ .

$y = - 10$

Step 2.7

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

$f (- 2) = - 2 {(- 2)}^{2} - 12 \cdot - 2 - 20$

Step 2.8

Simplify the result.

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

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

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Step 2.8.1.1.1

Multiply $- 2$ by ${(- 2)}^{2}$ .

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Step 2.8.1.1.1.1

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

$f (- 2) = (- 2) {(- 2)}^{2} - 12 \cdot - 2 - 20$

Step 2.8.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 2) = {(- 2)}^{1 + 2} - 12 \cdot - 2 - 20$

Step 2.8.1.1.2

Add $1$ and $2$ .

$f (- 2) = {(- 2)}^{3} - 12 \cdot - 2 - 20$

Step 2.8.1.2

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

$f (- 2) = - 8 - 12 \cdot - 2 - 20$

Step 2.8.1.3

Multiply $- 12$ by $- 2$ .

$f (- 2) = - 8 + 24 - 20$

Step 2.8.2

Simplify by adding and subtracting.

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Step 2.8.2.1

Add $- 8$ and $24$ .

$f (- 2) = 16 - 20$

Step 2.8.2.2

Subtract $20$ from $16$ .

$f (- 2) = - 4$

Step 2.8.3

The final answer is $- 4$ .

$- 4$

Step 2.9

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

$y = - 4$

Step 2.10

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

$f (- 1) = - 2 {(- 1)}^{2} - 12 \cdot - 1 - 20$

Step 2.11

Simplify the result.

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Step 2.11.1

Simplify each term.

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Step 2.11.1.1

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

$f (- 1) = - 2 \cdot 1 - 12 \cdot - 1 - 20$

Step 2.11.1.2

Multiply $- 2$ by $1$ .

$f (- 1) = - 2 - 12 \cdot - 1 - 20$

Step 2.11.1.3

Multiply $- 12$ by $- 1$ .

$f (- 1) = - 2 + 12 - 20$

Step 2.11.2

Simplify by adding and subtracting.

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Step 2.11.2.1

Add $- 2$ and $12$ .

$f (- 1) = 10 - 20$

Step 2.11.2.2

Subtract $20$ from $10$ .

$f (- 1) = - 10$

Step 2.11.3

The final answer is $- 10$ .

$- 10$

Step 2.12

The $y$ value at $x = - 1$ is $- 10$ .

$y = - 10$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 5 & - 10 \\ - 4 & - 4 \\ - 3 & - 2 \\ - 2 & - 4 \\ - 1 & - 10 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- 3, - 2)$

Focus: $(- 3, - \frac{17}{8})$

Axis of Symmetry: $x = - 3$

Directrix: $y = - \frac{15}{8}$

$\begin{array}{c} x & y \\ - 5 & - 10 \\ - 4 & - 4 \\ - 3 & - 2 \\ - 2 & - 4 \\ - 1 & - 10 \end{array}$

Step 4