Algebra Examples

$y = - 2 {(x - 2)}^{2} - 4$

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 $k$ .

$a = - 2$

$h = 2$

$k = - 4$

Step 1.2

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

Opens Down

Step 1.3

Find the vertex $(h, k)$ .

$(2, - 4)$

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.

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

Step 1.4.3

Simplify.

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Step 1.4.3.1

Multiply $4$ by $- 2$ .

$\frac{1}{- 8}$

Step 1.4.3.2

Move the negative in front of the fraction.

$- \frac{1}{8}$

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.

$(2, - \frac{33}{8})$

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.

$y = - \frac{31}{8}$

Step 1.8

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

Direction: Opens Down

Vertex: $(2, - 4)$

Focus: $(2, - \frac{33}{8})$

Axis of Symmetry: $x = 2$

Directrix: $y = - \frac{31}{8}$

Direction: Opens Down

Vertex: $(2, - 4)$

Focus: $(2, - \frac{33}{8})$

Axis of Symmetry: $x = 2$

Directrix: $y = - \frac{31}{8}$

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 $1$ in the expression.

$f (1) = - 2 {(1)}^{2} + 8 (1) - 12$

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

One to any power is one.

$f (1) = - 2 \cdot 1 + 8 (1) - 12$

Step 2.2.1.2

Multiply $- 2$ by $1$ .

$f (1) = - 2 + 8 (1) - 12$

Step 2.2.1.3

Multiply $8$ by $1$ .

$f (1) = - 2 + 8 - 12$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $- 2$ and $8$ .

$f (1) = 6 - 12$

Step 2.2.2.2

Subtract $12$ from $6$ .

$f (1) = - 6$

Step 2.2.3

The final answer is $- 6$ .

$- 6$

Step 2.3

The $y$ value at $x = 1$ is $- 6$ .

$y = - 6$

Step 2.4

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

$f (0) = - 2 {(0)}^{2} + 8 (0) - 12$

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

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

$f (0) = - 2 \cdot 0 + 8 (0) - 12$

Step 2.5.1.2

Multiply $- 2$ by $0$ .

$f (0) = 0 + 8 (0) - 12$

Step 2.5.1.3

Multiply $8$ by $0$ .

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

Step 2.5.2

Simplify by adding and subtracting.

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Step 2.5.2.1

Add $0$ and $0$ .

$f (0) = 0 - 12$

Step 2.5.2.2

Subtract $12$ from $0$ .

$f (0) = - 12$

Step 2.5.3

The final answer is $- 12$ .

$- 12$

Step 2.6

The $y$ value at $x = 0$ is $- 12$ .

$y = - 12$

Step 2.7

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

$f (3) = - 2 {(3)}^{2} + 8 (3) - 12$

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 $3$ to the power of $2$ .

$f (3) = - 2 \cdot 9 + 8 (3) - 12$

Step 2.8.1.2

Multiply $- 2$ by $9$ .

$f (3) = - 18 + 8 (3) - 12$

Step 2.8.1.3

Multiply $8$ by $3$ .

$f (3) = - 18 + 24 - 12$

Step 2.8.2

Simplify by adding and subtracting.

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Step 2.8.2.1

Add $- 18$ and $24$ .

$f (3) = 6 - 12$

Step 2.8.2.2

Subtract $12$ from $6$ .

$f (3) = - 6$

Step 2.8.3

The final answer is $- 6$ .

$- 6$

Step 2.9

The $y$ value at $x = 3$ is $- 6$ .

$y = - 6$

Step 2.10

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

$f (4) = - 2 {(4)}^{2} + 8 (4) - 12$

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 $4$ to the power of $2$ .

$f (4) = - 2 \cdot 16 + 8 (4) - 12$

Step 2.11.1.2

Multiply $- 2$ by $16$ .

$f (4) = - 32 + 8 (4) - 12$

Step 2.11.1.3

Multiply $8$ by $4$ .

$f (4) = - 32 + 32 - 12$

Step 2.11.2

Simplify by adding and subtracting.

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Step 2.11.2.1

Add $- 32$ and $32$ .

$f (4) = 0 - 12$

Step 2.11.2.2

Subtract $12$ from $0$ .

$f (4) = - 12$

Step 2.11.3

The final answer is $- 12$ .

$- 12$

Step 2.12

The $y$ value at $x = 4$ is $- 12$ .

$y = - 12$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & - 12 \\ 1 & - 6 \\ 2 & - 4 \\ 3 & - 6 \\ 4 & - 12 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(2, - 4)$

Focus: $(2, - \frac{33}{8})$

Axis of Symmetry: $x = 2$

Directrix: $y = - \frac{31}{8}$

$\begin{array}{c} x & y \\ 0 & - 12 \\ 1 & - 6 \\ 2 & - 4 \\ 3 & - 6 \\ 4 & - 12 \end{array}$

Step 4