Precalculus Examples

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

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 $k$ .

$a = - 1$

$h = 4$

$k = 0$

Step 1.2

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

Opens Down

Step 1.3

Find the vertex $(h, k)$ .

$(4, 0)$

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.

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

Step 1.4.3

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

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Step 1.4.3.1

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

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

Step 1.4.3.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

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.

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

Step 1.6

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

$x = 4$

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.

$y = \frac{1}{4}$

Step 1.8

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

Direction: Opens Down

Vertex: $(4, 0)$

Focus: $(4, - \frac{1}{4})$

Axis of Symmetry: $x = 4$

Directrix: $y = \frac{1}{4}$

Direction: Opens Down

Vertex: $(4, 0)$

Focus: $(4, - \frac{1}{4})$

Axis of Symmetry: $x = 4$

Directrix: $y = \frac{1}{4}$

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.

$f (3) = - {(3)}^{2} + 8 (3) - 16$

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 $2$ .

$f (3) = - 1 \cdot 9 + 8 (3) - 16$

Step 2.2.1.2

Multiply $- 1$ by $9$ .

$f (3) = - 9 + 8 (3) - 16$

Step 2.2.1.3

Multiply $8$ by $3$ .

$f (3) = - 9 + 24 - 16$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $- 9$ and $24$ .

$f (3) = 15 - 16$

Step 2.2.2.2

Subtract $16$ from $15$ .

$f (3) = - 1$

Step 2.2.3

The final answer is $- 1$ .

$- 1$

Step 2.3

The $y$ value at $x = 3$ is $- 1$ .

$y = - 1$

Step 2.4

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

$f (2) = - {(2)}^{2} + 8 (2) - 16$

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 $2$ to the power of $2$ .

$f (2) = - 1 \cdot 4 + 8 (2) - 16$

Step 2.5.1.2

Multiply $- 1$ by $4$ .

$f (2) = - 4 + 8 (2) - 16$

Step 2.5.1.3

Multiply $8$ by $2$ .

$f (2) = - 4 + 16 - 16$

Step 2.5.2

Simplify by adding and subtracting.

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Step 2.5.2.1

Add $- 4$ and $16$ .

$f (2) = 12 - 16$

Step 2.5.2.2

Subtract $16$ from $12$ .

$f (2) = - 4$

Step 2.5.3

The final answer is $- 4$ .

$- 4$

Step 2.6

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

$y = - 4$

Step 2.7

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

$f (5) = - {(5)}^{2} + 8 (5) - 16$

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 $5$ to the power of $2$ .

$f (5) = - 1 \cdot 25 + 8 (5) - 16$

Step 2.8.1.2

Multiply $- 1$ by $25$ .

$f (5) = - 25 + 8 (5) - 16$

Step 2.8.1.3

Multiply $8$ by $5$ .

$f (5) = - 25 + 40 - 16$

Step 2.8.2

Simplify by adding and subtracting.

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Step 2.8.2.1

Add $- 25$ and $40$ .

$f (5) = 15 - 16$

Step 2.8.2.2

Subtract $16$ from $15$ .

$f (5) = - 1$

Step 2.8.3

The final answer is $- 1$ .

$- 1$

Step 2.9

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

$y = - 1$

Step 2.10

Replace the variable $x$ with $6$ in the expression.

$f (6) = - {(6)}^{2} + 8 (6) - 16$

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 $6$ to the power of $2$ .

$f (6) = - 1 \cdot 36 + 8 (6) - 16$

Step 2.11.1.2

Multiply $- 1$ by $36$ .

$f (6) = - 36 + 8 (6) - 16$

Step 2.11.1.3

Multiply $8$ by $6$ .

$f (6) = - 36 + 48 - 16$

Step 2.11.2

Simplify by adding and subtracting.

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Step 2.11.2.1

Add $- 36$ and $48$ .

$f (6) = 12 - 16$

Step 2.11.2.2

Subtract $16$ from $12$ .

$f (6) = - 4$

Step 2.11.3

The final answer is $- 4$ .

$- 4$

Step 2.12

The $y$ value at $x = 6$ is $- 4$ .

$y = - 4$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 2 & - 4 \\ 3 & - 1 \\ 4 & 0 \\ 5 & - 1 \\ 6 & - 4 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(4, 0)$

Focus: $(4, - \frac{1}{4})$

Axis of Symmetry: $x = 4$

Directrix: $y = \frac{1}{4}$

$\begin{array}{c} x & y \\ 2 & - 4 \\ 3 & - 1 \\ 4 & 0 \\ 5 & - 1 \\ 6 & - 4 \end{array}$

Step 4