Trigonometry Examples

$h = - 16 t^{2} + 28 t + 6$

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

Complete the square for $- 16 x^{2} + 28 x + 6$ .

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Step 1.1.1.1

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

$a = - 16$

$b = 28$

$c = 6$

Step 1.1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

$d = \frac{28}{2 \cdot - 16}$

Step 1.1.1.3.2

Simplify the right side.

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Step 1.1.1.3.2.1

Cancel the common factor of $28$ and $2$ .

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Step 1.1.1.3.2.1.1

Factor $2$ out of $28$ .

$d = \frac{2 \cdot 14}{2 \cdot - 16}$

Step 1.1.1.3.2.1.2

Cancel the common factors.

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Step 1.1.1.3.2.1.2.1

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

$d = \frac{2 \cdot 14}{2 (- 16)}$

Step 1.1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 14}{2 \cdot - 16}$

Step 1.1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{14}{- 16}$

Step 1.1.1.3.2.2

Cancel the common factor of $14$ and $- 16$ .

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Step 1.1.1.3.2.2.1

Factor $2$ out of $14$ .

$d = \frac{2 (7)}{- 16}$

Step 1.1.1.3.2.2.2

Cancel the common factors.

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Step 1.1.1.3.2.2.2.1

Factor $2$ out of $- 16$ .

$d = \frac{2 \cdot 7}{2 \cdot - 8}$

Step 1.1.1.3.2.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 7}{2 \cdot - 8}$

Step 1.1.1.3.2.2.2.3

Rewrite the expression.

$d = \frac{7}{- 8}$

Step 1.1.1.3.2.3

Move the negative in front of the fraction.

$d = - \frac{7}{8}$

Step 1.1.1.4

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

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Step 1.1.1.4.1

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

$e = 6 - \frac{28^{2}}{4 \cdot - 16}$

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Simplify each term.

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Step 1.1.1.4.2.1.1

Raise $28$ to the power of $2$ .

$e = 6 - \frac{784}{4 \cdot - 16}$

Step 1.1.1.4.2.1.2

Multiply $4$ by $- 16$ .

$e = 6 - \frac{784}{- 64}$

Step 1.1.1.4.2.1.3

Cancel the common factor of $784$ and $- 64$ .

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Step 1.1.1.4.2.1.3.1

Factor $16$ out of $784$ .

$e = 6 - \frac{16 (49)}{- 64}$

Step 1.1.1.4.2.1.3.2

Cancel the common factors.

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Step 1.1.1.4.2.1.3.2.1

Factor $16$ out of $- 64$ .

$e = 6 - \frac{16 \cdot 49}{16 \cdot - 4}$

Step 1.1.1.4.2.1.3.2.2

Cancel the common factor.

$e = 6 - \frac{16 \cdot 49}{16 \cdot - 4}$

Step 1.1.1.4.2.1.3.2.3

Rewrite the expression.

$e = 6 - \frac{49}{- 4}$

Step 1.1.1.4.2.1.4

Move the negative in front of the fraction.

$e = 6 - - \frac{49}{4}$

Step 1.1.1.4.2.1.5

Multiply $- - \frac{49}{4}$ .

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Step 1.1.1.4.2.1.5.1

Multiply $- 1$ by $- 1$ .

$e = 6 + 1 (\frac{49}{4})$

Step 1.1.1.4.2.1.5.2

Multiply $\frac{49}{4}$ by $1$ .

$e = 6 + \frac{49}{4}$

Step 1.1.1.4.2.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 6 \cdot \frac{4}{4} + \frac{49}{4}$

Step 1.1.1.4.2.3

Combine $6$ and $\frac{4}{4}$ .

$e = \frac{6 \cdot 4}{4} + \frac{49}{4}$

Step 1.1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{6 \cdot 4 + 49}{4}$

Step 1.1.1.4.2.5

Simplify the numerator.

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Step 1.1.1.4.2.5.1

Multiply $6$ by $4$ .

$e = \frac{24 + 49}{4}$

Step 1.1.1.4.2.5.2

Add $24$ and $49$ .

$e = \frac{73}{4}$

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 16 {(x - \frac{7}{8})}^{2} + \frac{73}{4}$ .

$- 16 {(x - \frac{7}{8})}^{2} + \frac{73}{4}$

Step 1.1.2

Set $y$ equal to the new right side.

$y = - 16 {(x - \frac{7}{8})}^{2} + \frac{73}{4}$

Step 1.2

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

$a = - 16$

$h = \frac{7}{8}$

$k = \frac{73}{4}$

Step 1.3

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

Opens Down

Step 1.4

Find the vertex $(h, k)$ .

$(\frac{7}{8}, \frac{73}{4})$

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.

$(\frac{7}{8}, \frac{1167}{64})$

Step 1.7

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

$x = \frac{7}{8}$

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.

$y = \frac{1169}{64}$

Step 1.9

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

Direction: Opens Down

Vertex: $(\frac{7}{8}, \frac{73}{4})$

Focus: $(\frac{7}{8}, \frac{1167}{64})$

Axis of Symmetry: $x = \frac{7}{8}$

Directrix: $y = \frac{1169}{64}$

Direction: Opens Down

Vertex: $(\frac{7}{8}, \frac{73}{4})$

Focus: $(\frac{7}{8}, \frac{1167}{64})$

Axis of Symmetry: $x = \frac{7}{8}$

Directrix: $y = \frac{1169}{64}$

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 $0$ in the expression.

$f (0) = - 16 {(0)}^{2} + 28 (0) + 6$

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

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

$f (0) = - 16 \cdot 0 + 28 (0) + 6$

Step 2.2.1.2

Multiply $- 16$ by $0$ .

$f (0) = 0 + 28 (0) + 6$

Step 2.2.1.3

Multiply $28$ by $0$ .

$f (0) = 0 + 0 + 6$

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 6$

Step 2.2.2.2

Add $0$ and $6$ .

$f (0) = 6$

Step 2.2.3

The final answer is $6$ .

$6$

Step 2.3

The $y$ value at $x = 0$ is $6$ .

$y = 6$

Step 2.4

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

$f (2) = - 16 {(2)}^{2} + 28 (2) + 6$

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 $2$ .

$f (2) = - 16 \cdot 4 + 28 (2) + 6$

Step 2.5.1.2

Multiply $- 16$ by $4$ .

$f (2) = - 64 + 28 (2) + 6$

Step 2.5.1.3

Multiply $28$ by $2$ .

$f (2) = - 64 + 56 + 6$

Step 2.5.2

Simplify by adding numbers.

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Step 2.5.2.1

Add $- 64$ and $56$ .

$f (2) = - 8 + 6$

Step 2.5.2.2

Add $- 8$ and $6$ .

$f (2) = - 2$

Step 2.5.3

The final answer is $- 2$ .

$- 2$

Step 2.6

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

$y = - 2$

Step 2.7

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & 6 \\ \frac{7}{8} & \frac{73}{4} \\ 2 & - 2 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(\frac{7}{8}, \frac{73}{4})$

Focus: $(\frac{7}{8}, \frac{1167}{64})$

Axis of Symmetry: $x = \frac{7}{8}$

Directrix: $y = \frac{1169}{64}$

$\begin{array}{c} x & y \\ 0 & 6 \\ \frac{7}{8} & \frac{73}{4} \\ 2 & - 2 \end{array}$

Step 4