Precalculus Examples

$f (x) = 16 - \frac{1}{4} x^{2}$

Step 1

Find the properties of the given parabola.

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Step 1.1

Rewrite the equation in vertex form.

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Step 1.1.1

Simplify $16 - \frac{1}{4} x^{2}$ .

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Step 1.1.1.1

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

$y = 16 - \frac{x^{2}}{4}$

Step 1.1.1.2

Reorder $16$ and $- \frac{x^{2}}{4}$ .

$y = - \frac{x^{2}}{4} + 16$

Step 1.1.2

Complete the square for $- \frac{x^{2}}{4} + 16$ .

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Step 1.1.2.1

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

$a = - \frac{1}{4}$

$b = 0$

$c = 16$

Step 1.1.2.2

Consider the vertex form of a parabola.

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

Step 1.1.2.3

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

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Step 1.1.2.3.1

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

$d = \frac{0}{2 (- \frac{1}{4})}$

Step 1.1.2.3.2

Simplify the right side.

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Step 1.1.2.3.2.1

Cancel the common factor of $0$ and $2$ .

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Step 1.1.2.3.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (- \frac{1}{4})}$

Step 1.1.2.3.2.1.2

Cancel the common factors.

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Step 1.1.2.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (- \frac{1}{4})}$

Step 1.1.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{0}{- \frac{1}{4}}$

Step 1.1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (- 1 \cdot 4)$

Step 1.1.2.3.2.3

Multiply $0 (- 1 \cdot 4)$ .

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Step 1.1.2.3.2.3.1

Multiply $- 1$ by $4$ .

$d = 0 \cdot - 4$

Step 1.1.2.3.2.3.2

Multiply $0$ by $- 4$ .

$d = 0$

Step 1.1.2.4

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

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Step 1.1.2.4.1

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

$e = 16 - \frac{0^{2}}{4 (- \frac{1}{4})}$

Step 1.1.2.4.2

Simplify the right side.

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Step 1.1.2.4.2.1

Simplify each term.

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Step 1.1.2.4.2.1.1

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

$e = 16 - \frac{0}{4 (- \frac{1}{4})}$

Step 1.1.2.4.2.1.2

Simplify the denominator.

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Step 1.1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 16 - \frac{0}{- 4 (\frac{1}{4})}$

Step 1.1.2.4.2.1.2.2

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

$e = 16 - \frac{0}{\frac{- 4}{4}}$

Step 1.1.2.4.2.1.3

Divide $- 4$ by $4$ .

$e = 16 - \frac{0}{- 1}$

Step 1.1.2.4.2.1.4

Divide $0$ by $- 1$ .

$e = 16 - 0$

Step 1.1.2.4.2.1.5

Multiply $- 1$ by $0$ .

$e = 16 + 0$

Step 1.1.2.4.2.2

Add $16$ and $0$ .

$e = 16$

Step 1.1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{4} {(x + 0)}^{2} + 16$ .

$- \frac{1}{4} {(x + 0)}^{2} + 16$

Step 1.1.3

Set $y$ equal to the new right side.

$y = - \frac{1}{4} \cdot {(x + 0)}^{2} + 16$

Step 1.2

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

$a = - \frac{1}{4}$

$h = 0$

$k = 16$

Step 1.3

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

Opens Down

Step 1.4

Find the vertex $(h, k)$ .

$(0, 16)$

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.

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

Step 1.5.3

Simplify.

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Step 1.5.3.1

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

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Step 1.5.3.1.1

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

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

Step 1.5.3.1.2

Move the negative in front of the fraction.

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

Step 1.5.3.2

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

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

Step 1.5.3.3

Divide $4$ by $4$ .

$- \frac{1}{1}$

Step 1.5.3.4

Cancel the common factor of $1$ .

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Step 1.5.3.4.1

Cancel the common factor.

$- \frac{1}{1}$

Step 1.5.3.4.2

Rewrite the expression.

$- 1 \cdot 1$

Step 1.5.3.5

Multiply $- 1$ by $1$ .

$- 1$

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.

$(0, 15)$

Step 1.7

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

$x = 0$

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.

$y = 17$

Step 1.9

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

Direction: Opens Down

Vertex: $(0, 16)$

Focus: $(0, 15)$

Axis of Symmetry: $x = 0$

Directrix: $y = 17$

Direction: Opens Down

Vertex: $(0, 16)$

Focus: $(0, 15)$

Axis of Symmetry: $x = 0$

Directrix: $y = 17$

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 $- 2$ in the expression.

$f (- 2) = - \frac{{(- 2)}^{2}}{4} + 16$

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

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

Step 2.2.1.2

Cancel the common factor of $4$ .

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Step 2.2.1.2.1

Cancel the common factor.

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

Step 2.2.1.2.2

Rewrite the expression.

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

Step 2.2.1.3

Multiply $- 1$ by $1$ .

$f (- 2) = - 1 + 16$

Step 2.2.2

Add $- 1$ and $16$ .

$f (- 2) = 15$

Step 2.2.3

The final answer is $15$ .

$15$

Step 2.3

The $y$ value at $x = - 2$ is $15$ .

$y = 15$

Step 2.4

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

$f (- 1) = - \frac{{(- 1)}^{2}}{4} + 16$

Step 2.5

Simplify the result.

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Step 2.5.1

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

$f (- 1) = - \frac{1}{4} + 16$

Step 2.5.2

To write $16$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (- 1) = - \frac{1}{4} + 16 \cdot \frac{4}{4}$

Step 2.5.3

Combine $16$ and $\frac{4}{4}$ .

$f (- 1) = - \frac{1}{4} + \frac{16 \cdot 4}{4}$

Step 2.5.4

Combine the numerators over the common denominator.

$f (- 1) = \frac{- 1 + 16 \cdot 4}{4}$

Step 2.5.5

Simplify the numerator.

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Step 2.5.5.1

Multiply $16$ by $4$ .

$f (- 1) = \frac{- 1 + 64}{4}$

Step 2.5.5.2

Add $- 1$ and $64$ .

$f (- 1) = \frac{63}{4}$

Step 2.5.6

The final answer is $\frac{63}{4}$ .

$\frac{63}{4}$

Step 2.6

The $y$ value at $x = - 1$ is $\frac{63}{4}$ .

$y = \frac{63}{4}$

Step 2.7

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

$f (2) = - \frac{{(2)}^{2}}{4} + 16$

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 $2$ to the power of $2$ .

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

Step 2.8.1.2

Cancel the common factor of $4$ .

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Step 2.8.1.2.1

Cancel the common factor.

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

Step 2.8.1.2.2

Rewrite the expression.

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

Step 2.8.1.3

Multiply $- 1$ by $1$ .

$f (2) = - 1 + 16$

Step 2.8.2

Add $- 1$ and $16$ .

$f (2) = 15$

Step 2.8.3

The final answer is $15$ .

$15$

Step 2.9

The $y$ value at $x = 2$ is $15$ .

$y = 15$

Step 2.10

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

$f (1) = - \frac{{(1)}^{2}}{4} + 16$

Step 2.11

Simplify the result.

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Step 2.11.1

One to any power is one.

$f (1) = - \frac{1}{4} + 16$

Step 2.11.2

To write $16$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (1) = - \frac{1}{4} + 16 \cdot \frac{4}{4}$

Step 2.11.3

Combine $16$ and $\frac{4}{4}$ .

$f (1) = - \frac{1}{4} + \frac{16 \cdot 4}{4}$

Step 2.11.4

Combine the numerators over the common denominator.

$f (1) = \frac{- 1 + 16 \cdot 4}{4}$

Step 2.11.5

Simplify the numerator.

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Step 2.11.5.1

Multiply $16$ by $4$ .

$f (1) = \frac{- 1 + 64}{4}$

Step 2.11.5.2

Add $- 1$ and $64$ .

$f (1) = \frac{63}{4}$

Step 2.11.6

The final answer is $\frac{63}{4}$ .

$\frac{63}{4}$

Step 2.12

The $y$ value at $x = 1$ is $\frac{63}{4}$ .

$y = \frac{63}{4}$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 15 \\ - 1 & \frac{63}{4} \\ 0 & 16 \\ 1 & \frac{63}{4} \\ 2 & 15 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(0, 16)$

Focus: $(0, 15)$

Axis of Symmetry: $x = 0$

Directrix: $y = 17$

$\begin{array}{c} x & y \\ - 2 & 15 \\ - 1 & \frac{63}{4} \\ 0 & 16 \\ 1 & \frac{63}{4} \\ 2 & 15 \end{array}$

Step 4