Precalculus Examples

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

Step 1

Find the properties of the given parabola.

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Step 1.1

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

$a = 3$

$h = - 2$

$k = - 7$

Step 1.2

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

Opens Up

Step 1.3

Find the vertex $(h, k)$ .

$(- 2, - 7)$

Step 1.4

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

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Step 1.4.1

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

$\frac{1}{4 a}$

Step 1.4.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot 3}$

Step 1.4.3

Multiply $4$ by $3$ .

$\frac{1}{12}$

Step 1.5

Find the focus.

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Step 1.5.1

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 1.5.2

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

$(- 2, - \frac{83}{12})$

Step 1.6

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

$x = - 2$

Step 1.7

Find the directrix.

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Step 1.7.1

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 1.7.2

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

$y = - \frac{85}{12}$

Step 1.8

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

Direction: Opens Up

Vertex: $(- 2, - 7)$

Focus: $(- 2, - \frac{83}{12})$

Axis of Symmetry: $x = - 2$

Directrix: $y = - \frac{85}{12}$

Direction: Opens Up

Vertex: $(- 2, - 7)$

Focus: $(- 2, - \frac{83}{12})$

Axis of Symmetry: $x = - 2$

Directrix: $y = - \frac{85}{12}$

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.

$f (- 3) = 3 {(- 3)}^{2} + 12 (- 3) + 5$

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

$f (- 3) = 3 \cdot 9 + 12 (- 3) + 5$

Step 2.2.1.2

Multiply $3$ by $9$ .

$f (- 3) = 27 + 12 (- 3) + 5$

Step 2.2.1.3

Multiply $12$ by $- 3$ .

$f (- 3) = 27 - 36 + 5$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $36$ from $27$ .

$f (- 3) = - 9 + 5$

Step 2.2.2.2

Add $- 9$ and $5$ .

$f (- 3) = - 4$

Step 2.2.3

The final answer is $- 4$ .

$- 4$

Step 2.3

The $y$ value at $x = - 3$ is $- 4$ .

$y = - 4$

Step 2.4

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

$f (- 4) = 3 {(- 4)}^{2} + 12 (- 4) + 5$

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

$f (- 4) = 3 \cdot 16 + 12 (- 4) + 5$

Step 2.5.1.2

Multiply $3$ by $16$ .

$f (- 4) = 48 + 12 (- 4) + 5$

Step 2.5.1.3

Multiply $12$ by $- 4$ .

$f (- 4) = 48 - 48 + 5$

Step 2.5.2

Simplify by adding and subtracting.

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Step 2.5.2.1

Subtract $48$ from $48$ .

$f (- 4) = 0 + 5$

Step 2.5.2.2

Add $0$ and $5$ .

$f (- 4) = 5$

Step 2.5.3

The final answer is $5$ .

$5$

Step 2.6

The $y$ value at $x = - 4$ is $5$ .

$y = 5$

Step 2.7

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

$f (- 1) = 3 {(- 1)}^{2} + 12 (- 1) + 5$

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

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

$f (- 1) = 3 \cdot 1 + 12 (- 1) + 5$

Step 2.8.1.2

Multiply $3$ by $1$ .

$f (- 1) = 3 + 12 (- 1) + 5$

Step 2.8.1.3

Multiply $12$ by $- 1$ .

$f (- 1) = 3 - 12 + 5$

Step 2.8.2

Simplify by adding and subtracting.

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Step 2.8.2.1

Subtract $12$ from $3$ .

$f (- 1) = - 9 + 5$

Step 2.8.2.2

Add $- 9$ and $5$ .

$f (- 1) = - 4$

Step 2.8.3

The final answer is $- 4$ .

$- 4$

Step 2.9

The $y$ value at $x = - 1$ is $- 4$ .

$y = - 4$

Step 2.10

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

$f (0) = 3 {(0)}^{2} + 12 (0) + 5$

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

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

$f (0) = 3 \cdot 0 + 12 (0) + 5$

Step 2.11.1.2

Multiply $3$ by $0$ .

$f (0) = 0 + 12 (0) + 5$

Step 2.11.1.3

Multiply $12$ by $0$ .

$f (0) = 0 + 0 + 5$

Step 2.11.2

Simplify by adding numbers.

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Step 2.11.2.1

Add $0$ and $0$ .

$f (0) = 0 + 5$

Step 2.11.2.2

Add $0$ and $5$ .

$f (0) = 5$

Step 2.11.3

The final answer is $5$ .

$5$

Step 2.12

The $y$ value at $x = 0$ is $5$ .

$y = 5$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 4 & 5 \\ - 3 & - 4 \\ - 2 & - 7 \\ - 1 & - 4 \\ 0 & 5 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 2, - 7)$

Focus: $(- 2, - \frac{83}{12})$

Axis of Symmetry: $x = - 2$

Directrix: $y = - \frac{85}{12}$

$\begin{array}{c} x & y \\ - 4 & 5 \\ - 3 & - 4 \\ - 2 & - 7 \\ - 1 & - 4 \\ 0 & 5 \end{array}$

Step 4