Trigonometry Examples

$y = \frac{1}{2} \cdot ((x - 5) (x + 2))$

Step 1

Solve for $y$ .

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Step 1.1

Remove parentheses.

$y = \frac{1}{2} \cdot ((x - 5) (x + 2))$

Step 1.2

Simplify $\frac{1}{2} \cdot ((x - 5) (x + 2))$ .

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Step 1.2.1

Expand $(x - 5) (x + 2)$ using the FOIL Method.

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Step 1.2.1.1

Apply the distributive property.

$y = \frac{1}{2} \cdot (x (x + 2) - 5 (x + 2))$

Step 1.2.1.2

Apply the distributive property.

$y = \frac{1}{2} \cdot (x \cdot x + x \cdot 2 - 5 (x + 2))$

Step 1.2.1.3

Apply the distributive property.

$y = \frac{1}{2} \cdot (x \cdot x + x \cdot 2 - 5 x - 5 \cdot 2)$

Step 1.2.2

Simplify and combine like terms.

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

Multiply $x$ by $x$ .

$y = \frac{1}{2} \cdot (x^{2} + x \cdot 2 - 5 x - 5 \cdot 2)$

Step 1.2.2.1.2

Move $2$ to the left of $x$ .

$y = \frac{1}{2} \cdot (x^{2} + 2 \cdot x - 5 x - 5 \cdot 2)$

Step 1.2.2.1.3

Multiply $- 5$ by $2$ .

$y = \frac{1}{2} \cdot (x^{2} + 2 x - 5 x - 10)$

Step 1.2.2.2

Subtract $5 x$ from $2 x$ .

$y = \frac{1}{2} \cdot (x^{2} - 3 x - 10)$

Step 1.2.3

Apply the distributive property.

$y = \frac{1}{2} x^{2} + \frac{1}{2} (- 3 x) + \frac{1}{2} \cdot - 10$

Step 1.2.4

Simplify.

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Step 1.2.4.1

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

$y = \frac{x^{2}}{2} + \frac{1}{2} (- 3 x) + \frac{1}{2} \cdot - 10$

Step 1.2.4.2

Multiply $\frac{1}{2} (- 3 x)$ .

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Step 1.2.4.2.1

Combine $- 3$ and $\frac{1}{2}$ .

$y = \frac{x^{2}}{2} + \frac{- 3}{2} x + \frac{1}{2} \cdot - 10$

Step 1.2.4.2.2

Combine $\frac{- 3}{2}$ and $x$ .

$y = \frac{x^{2}}{2} + \frac{- 3 x}{2} + \frac{1}{2} \cdot - 10$

Step 1.2.4.3

Cancel the common factor of $2$ .

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Step 1.2.4.3.1

Factor $2$ out of $- 10$ .

$y = \frac{x^{2}}{2} + \frac{- 3 x}{2} + \frac{1}{2} \cdot (2 (- 5))$

Step 1.2.4.3.2

Cancel the common factor.

$y = \frac{x^{2}}{2} + \frac{- 3 x}{2} + \frac{1}{2} \cdot (2 \cdot - 5)$

Step 1.2.4.3.3

Rewrite the expression.

$y = \frac{x^{2}}{2} + \frac{- 3 x}{2} - 5$

Step 1.2.5

Move the negative in front of the fraction.

$y = \frac{x^{2}}{2} - \frac{3 x}{2} - 5$

Step 2

Find the properties of the given parabola.

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Step 2.1

Rewrite the equation in vertex form.

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Step 2.1.1

Complete the square for $\frac{x^{2}}{2} - \frac{3 x}{2} - 5$ .

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Step 2.1.1.1

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

$a = \frac{1}{2}$

$b = - \frac{3}{2}$

$c = - 5$

Step 2.1.1.2

Consider the vertex form of a parabola.

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

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

$d = \frac{- \frac{3}{2}}{2 (\frac{1}{2})}$

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{3}{2} \cdot \frac{1}{2 (\frac{1}{2})}$

Step 2.1.1.3.2.2

Combine $2$ and $\frac{1}{2}$ .

$d = - \frac{3}{2} \cdot \frac{1}{\frac{2}{2}}$

Step 2.1.1.3.2.3

Divide $2$ by $2$ .

$d = - \frac{3}{2} \cdot \frac{1}{1}$

Step 2.1.1.3.2.4

Cancel the common factor of $1$ .

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Step 2.1.1.3.2.4.1

Cancel the common factor.

$d = - \frac{3}{2} \cdot \frac{1}{1}$

Step 2.1.1.3.2.4.2

Rewrite the expression.

$d = - \frac{3}{2} \cdot 1$

Step 2.1.1.3.2.5

Multiply $- \frac{3}{2}$ by $1$ .

$d = - \frac{3}{2}$

Step 2.1.1.4

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

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Step 2.1.1.4.1

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

$e = - 5 - \frac{{(- \frac{3}{2})}^{2}}{4 (\frac{1}{2})}$

Step 2.1.1.4.2

Simplify the right side.

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Step 2.1.1.4.2.1

Simplify each term.

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Step 2.1.1.4.2.1.1

Simplify the numerator.

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Step 2.1.1.4.2.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

$e = - 5 - \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2}}{4 (\frac{1}{2})}$

Step 2.1.1.4.2.1.1.2

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

$e = - 5 - \frac{1 {(\frac{3}{2})}^{2}}{4 (\frac{1}{2})}$

Step 2.1.1.4.2.1.1.3

Apply the product rule to $\frac{3}{2}$ .

$e = - 5 - \frac{1 \frac{3^{2}}{2^{2}}}{4 (\frac{1}{2})}$

Step 2.1.1.4.2.1.1.4

Raise $3$ to the power of $2$ .

$e = - 5 - \frac{1 \frac{9}{2^{2}}}{4 (\frac{1}{2})}$

Step 2.1.1.4.2.1.1.5

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

$e = - 5 - \frac{1 (\frac{9}{4})}{4 (\frac{1}{2})}$

Step 2.1.1.4.2.1.1.6

Multiply $\frac{9}{4}$ by $1$ .

$e = - 5 - \frac{\frac{9}{4}}{4 (\frac{1}{2})}$

Step 2.1.1.4.2.1.2

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

$e = - 5 - \frac{\frac{9}{4}}{\frac{4}{2}}$

Step 2.1.1.4.2.1.3

Divide $4$ by $2$ .

$e = - 5 - \frac{\frac{9}{4}}{2}$

Step 2.1.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - 5 - (\frac{9}{4} \cdot \frac{1}{2})$

Step 2.1.1.4.2.1.5

Multiply $\frac{9}{4} \cdot \frac{1}{2}$ .

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Step 2.1.1.4.2.1.5.1

Multiply $\frac{9}{4}$ by $\frac{1}{2}$ .

$e = - 5 - \frac{9}{4 \cdot 2}$

Step 2.1.1.4.2.1.5.2

Multiply $4$ by $2$ .

$e = - 5 - \frac{9}{8}$

Step 2.1.1.4.2.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$e = - 5 \cdot \frac{8}{8} - \frac{9}{8}$

Step 2.1.1.4.2.3

Combine $- 5$ and $\frac{8}{8}$ .

$e = \frac{- 5 \cdot 8}{8} - \frac{9}{8}$

Step 2.1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{- 5 \cdot 8 - 9}{8}$

Step 2.1.1.4.2.5

Simplify the numerator.

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Step 2.1.1.4.2.5.1

Multiply $- 5$ by $8$ .

$e = \frac{- 40 - 9}{8}$

Step 2.1.1.4.2.5.2

Subtract $9$ from $- 40$ .

$e = \frac{- 49}{8}$

Step 2.1.1.4.2.6

Move the negative in front of the fraction.

$e = - \frac{49}{8}$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(x - \frac{3}{2})}^{2} - \frac{49}{8}$ .

$\frac{1}{2} {(x - \frac{3}{2})}^{2} - \frac{49}{8}$

Step 2.1.2

Set $y$ equal to the new right side.

$y = \frac{1}{2} \cdot {(x - \frac{3}{2})}^{2} - \frac{49}{8}$

Step 2.2

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

$a = \frac{1}{2}$

$h = \frac{3}{2}$

$k = - \frac{49}{8}$

Step 2.3

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Step 2.4

Find the vertex $(h, k)$ .

$(\frac{3}{2}, - \frac{49}{8})$

Step 2.5

Find $p$ , the distance from the vertex to the focus.

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Step 2.5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 2.5.2

Substitute the value of $a$ into the formula.

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

Step 2.5.3

Simplify.

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Step 2.5.3.1

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

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

Step 2.5.3.2

Divide $4$ by $2$ .

$\frac{1}{2}$

Step 2.6

Find the focus.

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Step 2.6.1

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 2.6.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(\frac{3}{2}, - \frac{45}{8})$

Step 2.7

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

$x = \frac{3}{2}$

Step 2.8

Find the directrix.

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Step 2.8.1

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 2.8.2

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = - \frac{53}{8}$

Step 2.9

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

Direction: Opens Up

Vertex: $(\frac{3}{2}, - \frac{49}{8})$

Focus: $(\frac{3}{2}, - \frac{45}{8})$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{53}{8}$

Direction: Opens Up

Vertex: $(\frac{3}{2}, - \frac{49}{8})$

Focus: $(\frac{3}{2}, - \frac{45}{8})$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{53}{8}$

Step 3

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 3.1

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

$f (1) = \frac{{(1)}^{2}}{2} - \frac{3 (1)}{2} - 5$

Step 3.2

Simplify the result.

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Step 3.2.1

Combine the numerators over the common denominator.

$f (1) = - 5 + \frac{{(1)}^{2} - 3 \cdot 1}{2}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

One to any power is one.

$f (1) = - 5 + \frac{1 - 3 \cdot 1}{2}$

Step 3.2.2.2

Multiply $- 3$ by $1$ .

$f (1) = - 5 + \frac{1 - 3}{2}$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Subtract $3$ from $1$ .

$f (1) = - 5 + \frac{- 2}{2}$

Step 3.2.3.2

Divide $- 2$ by $2$ .

$f (1) = - 5 - 1$

Step 3.2.3.3

Subtract $1$ from $- 5$ .

$f (1) = - 6$

Step 3.2.4

The final answer is $- 6$ .

$- 6$

Step 3.3

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

$y = - 6$

Step 3.4

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

$f (0) = \frac{{(0)}^{2}}{2} - \frac{3 (0)}{2} - 5$

Step 3.5

Simplify the result.

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Step 3.5.1

Combine the numerators over the common denominator.

$f (0) = - 5 + \frac{{(0)}^{2} - 3 \cdot 0}{2}$

Step 3.5.2

Simplify each term.

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Step 3.5.2.1

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

$f (0) = - 5 + \frac{0 - 3 \cdot 0}{2}$

Step 3.5.2.2

Multiply $- 3$ by $0$ .

$f (0) = - 5 + \frac{0 + 0}{2}$

Step 3.5.3

Simplify the expression.

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Step 3.5.3.1

Add $0$ and $0$ .

$f (0) = - 5 + \frac{0}{2}$

Step 3.5.3.2

Divide $0$ by $2$ .

$f (0) = - 5 + 0$

Step 3.5.3.3

Add $- 5$ and $0$ .

$f (0) = - 5$

Step 3.5.4

The final answer is $- 5$ .

$- 5$

Step 3.6

The $y$ value at $x = 0$ is $- 5$ .

$y = - 5$

Step 3.7

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

$f (3) = \frac{{(3)}^{2}}{2} - \frac{3 (3)}{2} - 5$

Step 3.8

Simplify the result.

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Step 3.8.1

Combine the numerators over the common denominator.

$f (3) = - 5 + \frac{{(3)}^{2} - 3 \cdot 3}{2}$

Step 3.8.2

Simplify each term.

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Step 3.8.2.1

Raise $3$ to the power of $2$ .

$f (3) = - 5 + \frac{9 - 3 \cdot 3}{2}$

Step 3.8.2.2

Multiply $- 3$ by $3$ .

$f (3) = - 5 + \frac{9 - 9}{2}$

Step 3.8.3

Simplify the expression.

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Step 3.8.3.1

Subtract $9$ from $9$ .

$f (3) = - 5 + \frac{0}{2}$

Step 3.8.3.2

Divide $0$ by $2$ .

$f (3) = - 5 + 0$

Step 3.8.3.3

Add $- 5$ and $0$ .

$f (3) = - 5$

Step 3.8.4

The final answer is $- 5$ .

$- 5$

Step 3.9

The $y$ value at $x = 3$ is $- 5$ .

$y = - 5$

Step 3.10

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

$f (4) = \frac{{(4)}^{2}}{2} - \frac{3 (4)}{2} - 5$

Step 3.11

Simplify the result.

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Step 3.11.1

Combine the numerators over the common denominator.

$f (4) = - 5 + \frac{{(4)}^{2} - 3 \cdot 4}{2}$

Step 3.11.2

Simplify each term.

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Step 3.11.2.1

Raise $4$ to the power of $2$ .

$f (4) = - 5 + \frac{16 - 3 \cdot 4}{2}$

Step 3.11.2.2

Multiply $- 3$ by $4$ .

$f (4) = - 5 + \frac{16 - 12}{2}$

Step 3.11.3

Simplify the expression.

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Step 3.11.3.1

Subtract $12$ from $16$ .

$f (4) = - 5 + \frac{4}{2}$

Step 3.11.3.2

Divide $4$ by $2$ .

$f (4) = - 5 + 2$

Step 3.11.3.3

Add $- 5$ and $2$ .

$f (4) = - 3$

Step 3.11.4

The final answer is $- 3$ .

$- 3$

Step 3.12

The $y$ value at $x = 4$ is $- 3$ .

$y = - 3$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & - 5 \\ 1 & - 6 \\ \frac{3}{2} & - \frac{49}{8} \\ 3 & - 5 \\ 4 & - 3 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(\frac{3}{2}, - \frac{49}{8})$

Focus: $(\frac{3}{2}, - \frac{45}{8})$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{53}{8}$

$\begin{array}{c} x & y \\ 0 & - 5 \\ 1 & - 6 \\ \frac{3}{2} & - \frac{49}{8} \\ 3 & - 5 \\ 4 & - 3 \end{array}$

Step 5