Trigonometry Examples

$y = 176 x - 16 x^{2}$

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

Reorder $176 x$ and $- 16 x^{2}$ .

$y = - 16 x^{2} + 176 x$

Step 1.1.2

Complete the square for $- 16 x^{2} + 176 x$ .

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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 $c$ .

$a = - 16$

$b = 176$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{176}{2 \cdot - 16}$

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 $176$ and $2$ .

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Step 1.1.2.3.2.1.1

Factor $2$ out of $176$ .

$d = \frac{2 \cdot 88}{2 \cdot - 16}$

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

Factor $2$ out of $2 \cdot - 16$ .

$d = \frac{2 \cdot 88}{2 (- 16)}$

Step 1.1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 88}{2 \cdot - 16}$

Step 1.1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{88}{- 16}$

Step 1.1.2.3.2.2

Cancel the common factor of $88$ and $- 16$ .

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Step 1.1.2.3.2.2.1

Factor $8$ out of $88$ .

$d = \frac{8 (11)}{- 16}$

Step 1.1.2.3.2.2.2

Cancel the common factors.

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Step 1.1.2.3.2.2.2.1

Factor $8$ out of $- 16$ .

$d = \frac{8 \cdot 11}{8 \cdot - 2}$

Step 1.1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{8 \cdot 11}{8 \cdot - 2}$

Step 1.1.2.3.2.2.2.3

Rewrite the expression.

$d = \frac{11}{- 2}$

Step 1.1.2.3.2.3

Move the negative in front of the fraction.

$d = - \frac{11}{2}$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{176^{2}}{4 \cdot - 16}$

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

Raise $176$ to the power of $2$ .

$e = 0 - \frac{30976}{4 \cdot - 16}$

Step 1.1.2.4.2.1.2

Multiply $4$ by $- 16$ .

$e = 0 - \frac{30976}{- 64}$

Step 1.1.2.4.2.1.3

Divide $30976$ by $- 64$ .

$e = 0 - - 484$

Step 1.1.2.4.2.1.4

Multiply $- 1$ by $- 484$ .

$e = 0 + 484$

Step 1.1.2.4.2.2

Add $0$ and $484$ .

$e = 484$

Step 1.1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 16 {(x - \frac{11}{2})}^{2} + 484$ .

$- 16 {(x - \frac{11}{2})}^{2} + 484$

Step 1.1.3

Set $y$ equal to the new right side.

$y = - 16 {(x - \frac{11}{2})}^{2} + 484$

Step 1.2

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

$a = - 16$

$h = \frac{11}{2}$

$k = 484$

Step 1.3

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

Opens Down

Step 1.4

Find the vertex $(h, k)$ .

$(\frac{11}{2}, 484)$

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.

$\frac{1}{4 \cdot - 16}$

Step 1.5.3

Simplify.

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Step 1.5.3.1

Multiply $4$ by $- 16$ .

$\frac{1}{- 64}$

Step 1.5.3.2

Move the negative in front of the fraction.

$- \frac{1}{64}$

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.

$(\frac{11}{2}, \frac{30975}{64})$

Step 1.7

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

$x = \frac{11}{2}$

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.

$y = \frac{30977}{64}$

Step 1.9

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

Direction: Opens Down

Vertex: $(\frac{11}{2}, 484)$

Focus: $(\frac{11}{2}, \frac{30975}{64})$

Axis of Symmetry: $x = \frac{11}{2}$

Directrix: $y = \frac{30977}{64}$

Direction: Opens Down

Vertex: $(\frac{11}{2}, 484)$

Focus: $(\frac{11}{2}, \frac{30975}{64})$

Axis of Symmetry: $x = \frac{11}{2}$

Directrix: $y = \frac{30977}{64}$

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 $5$ in the expression.

$f (5) = - 16 {(5)}^{2} + 176 (5)$

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 $5$ to the power of $2$ .

$f (5) = - 16 \cdot 25 + 176 (5)$

Step 2.2.1.2

Multiply $- 16$ by $25$ .

$f (5) = - 400 + 176 (5)$

Step 2.2.1.3

Multiply $176$ by $5$ .

$f (5) = - 400 + 880$

Step 2.2.2

Add $- 400$ and $880$ .

$f (5) = 480$

Step 2.2.3

The final answer is $480$ .

$480$

Step 2.3

The $y$ value at $x = 5$ is $480$ .

$y = 480$

Step 2.4

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 5 & 480 \\ \frac{11}{2} & 484 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(\frac{11}{2}, 484)$

Focus: $(\frac{11}{2}, \frac{30975}{64})$

Axis of Symmetry: $x = \frac{11}{2}$

Directrix: $y = \frac{30977}{64}$

$\begin{array}{c} x & y \\ 5 & 480 \\ \frac{11}{2} & 484 \end{array}$

Step 4