Trigonometry Examples

$f (x) = - 12 + 9 \cos (\frac{π}{4}) (x - 3)$

Step 1

Rewrite the function as an equation.

$y = - 12 + \frac{9 \sqrt{2} x}{2} - \frac{27 \sqrt{2}}{2}$

Step 2

Reorder $- 12$ and $\frac{9 \sqrt{2} x}{2}$ .

$y = \frac{9 \sqrt{2} x}{2} - 12 - \frac{27 \sqrt{2}}{2}$

Step 3

Rewrite in slope-intercept form.

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Step 3.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 3.2

Write in $y = m x + b$ form.

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Step 3.2.1

Rearrange terms.

$y = \frac{9 \sqrt{2} x}{2} - 12 - \frac{27 \sqrt{2}}{2}$

Step 3.2.2

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{9 \sqrt{2} x}{2} - 12 \cdot \frac{2}{2} - \frac{27 \sqrt{2}}{2}$

Step 3.2.3

Combine $- 12$ and $\frac{2}{2}$ .

$y = \frac{9 \sqrt{2} x}{2} + \frac{- 12 \cdot 2}{2} - \frac{27 \sqrt{2}}{2}$

Step 3.2.4

Combine the numerators over the common denominator.

$y = \frac{9 \sqrt{2} x}{2} + \frac{- 12 \cdot 2 - 27 \sqrt{2}}{2}$

Step 3.2.5

Multiply $- 12$ by $2$ .

$y = \frac{9 \sqrt{2} x}{2} + \frac{- 24 - 27 \sqrt{2}}{2}$

Step 3.2.6

Rewrite $- 24$ as $- 1 (24)$ .

$y = \frac{9 \sqrt{2} x}{2} + \frac{- 1 (24) - 27 \sqrt{2}}{2}$

Step 3.2.7

Factor $- 1$ out of $- 27 \sqrt{2}$ .

$y = \frac{9 \sqrt{2} x}{2} + \frac{- 1 (24) - (27 \sqrt{2})}{2}$

Step 3.2.8

Factor $- 1$ out of $- 1 (24) - (27 \sqrt{2})$ .

$y = \frac{9 \sqrt{2} x}{2} + \frac{- 1 (24 + 27 \sqrt{2})}{2}$

Step 3.2.9

Move the negative in front of the fraction.

$y = \frac{9 \sqrt{2} x}{2} - \frac{24 + 27 \sqrt{2}}{2}$

Step 3.2.10

Reorder terms.

$y = \frac{9 \sqrt{2}}{2} x - \frac{24 + 27 \sqrt{2}}{2}$

Step 4

Use the slope-intercept form to find the slope and y-intercept.

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Step 4.1

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m = \frac{9}{2}$

$b = - \frac{24 + 27 \sqrt{2}}{2}$

Step 4.2

The slope of the line is the value of $m$ , and the y-intercept is the value of $b$ .

Slope: $\frac{9}{2}$

y-intercept: $(0, - \frac{24 + 27 \sqrt{2}}{2})$

Slope: $\frac{9}{2}$

y-intercept: $(0, - \frac{24 + 27 \sqrt{2}}{2})$

Step 5

Any line can be graphed using two points. Select two $x$ values, and plug them into the equation to find the corresponding $y$ values.

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Step 5.1

Write in $y = m x + b$ form.

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Step 5.1.1

Reorder $- 12$ and $\frac{9 \sqrt{2} x}{2}$ .

$y = \frac{9 \sqrt{2} x}{2} - 12 - \frac{27 \sqrt{2}}{2}$

Step 5.1.2

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{9 \sqrt{2} x}{2} - 12 \cdot \frac{2}{2} - \frac{27 \sqrt{2}}{2}$

Step 5.1.3

Combine $- 12$ and $\frac{2}{2}$ .

$y = \frac{9 \sqrt{2} x}{2} + \frac{- 12 \cdot 2}{2} - \frac{27 \sqrt{2}}{2}$

Step 5.1.4

Combine the numerators over the common denominator.

$y = \frac{9 \sqrt{2} x}{2} + \frac{- 12 \cdot 2 - 27 \sqrt{2}}{2}$

Step 5.1.5

Multiply $- 12$ by $2$ .

$y = \frac{9 \sqrt{2} x}{2} + \frac{- 24 - 27 \sqrt{2}}{2}$

Step 5.1.6

Rewrite $- 24$ as $- 1 (24)$ .

$y = \frac{9 \sqrt{2} x}{2} + \frac{- 1 (24) - 27 \sqrt{2}}{2}$

Step 5.1.7

Factor $- 1$ out of $- 27 \sqrt{2}$ .

$y = \frac{9 \sqrt{2} x}{2} + \frac{- 1 (24) - (27 \sqrt{2})}{2}$

Step 5.1.8

Factor $- 1$ out of $- 1 (24) - (27 \sqrt{2})$ .

$y = \frac{9 \sqrt{2} x}{2} + \frac{- 1 (24 + 27 \sqrt{2})}{2}$

Step 5.1.9

Move the negative in front of the fraction.

$y = \frac{9 \sqrt{2} x}{2} - \frac{24 + 27 \sqrt{2}}{2}$

Step 5.1.10

Reorder terms.

$y = \frac{9 \sqrt{2}}{2} x - \frac{24 + 27 \sqrt{2}}{2}$

Step 5.2

Find the x-intercept.

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Step 5.2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{9 \sqrt{2}}{2} x - \frac{24 + 27 \sqrt{2}}{2}$

Step 5.2.2

Solve the equation.

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Step 5.2.2.1

Rewrite the equation as $\frac{9 \sqrt{2}}{2} x - \frac{24 + 27 \sqrt{2}}{2} = 0$ .

$\frac{9 \sqrt{2}}{2} x - \frac{24 + 27 \sqrt{2}}{2} = 0$

Step 5.2.2.2

Simplify $\frac{9 \sqrt{2}}{2} x - \frac{24 + 27 \sqrt{2}}{2}$ .

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Step 5.2.2.2.1

Simplify terms.

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Step 5.2.2.2.1.1

Combine $\frac{9 \sqrt{2}}{2}$ and $x$ .

$\frac{9 \sqrt{2} x}{2} - \frac{24 + 27 \sqrt{2}}{2} = 0$

Step 5.2.2.2.1.2

Combine the numerators over the common denominator.

$\frac{9 \sqrt{2} x - (24 + 27 \sqrt{2})}{2} = 0$

Step 5.2.2.2.2

Simplify each term.

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Step 5.2.2.2.2.1

Apply the distributive property.

$\frac{9 \sqrt{2} x - 1 \cdot 24 - (27 \sqrt{2})}{2} = 0$

Step 5.2.2.2.2.2

Multiply $- 1$ by $24$ .

$\frac{9 \sqrt{2} x - 24 - (27 \sqrt{2})}{2} = 0$

Step 5.2.2.2.2.3

Multiply $27$ by $- 1$ .

$\frac{9 \sqrt{2} x - 24 - 27 \sqrt{2}}{2} = 0$

Step 5.2.2.2.3

Factor $3$ out of $9 \sqrt{2} x - 24 - 27 \sqrt{2}$ .

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Step 5.2.2.2.3.1

Factor $3$ out of $9 \sqrt{2} x$ .

$\frac{3 (3 \sqrt{2} x) - 24 - 27 \sqrt{2}}{2} = 0$

Step 5.2.2.2.3.2

Factor $3$ out of $- 24$ .

$\frac{3 (3 \sqrt{2} x) + 3 (- 8) - 27 \sqrt{2}}{2} = 0$

Step 5.2.2.2.3.3

Factor $3$ out of $- 27 \sqrt{2}$ .

$\frac{3 (3 \sqrt{2} x) + 3 (- 8) + 3 (- 9 \sqrt{2})}{2} = 0$

Step 5.2.2.2.3.4

Factor $3$ out of $3 (3 \sqrt{2} x) + 3 (- 8)$ .

$\frac{3 (3 \sqrt{2} x - 8) + 3 (- 9 \sqrt{2})}{2} = 0$

Step 5.2.2.2.3.5

Factor $3$ out of $3 (3 \sqrt{2} x - 8) + 3 (- 9 \sqrt{2})$ .

$\frac{3 (3 \sqrt{2} x - 8 - 9 \sqrt{2})}{2} = 0$

Step 5.2.2.3

Set the numerator equal to zero.

$3 (3 \sqrt{2} x - 8 - 9 \sqrt{2}) = 0$

Step 5.2.2.4

Solve the equation for $x$ .

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Step 5.2.2.4.1

Divide each term in $3 (3 \sqrt{2} x - 8 - 9 \sqrt{2}) = 0$ by $3$ and simplify.

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Step 5.2.2.4.1.1

Divide each term in $3 (3 \sqrt{2} x - 8 - 9 \sqrt{2}) = 0$ by $3$ .

$\frac{3 (3 \sqrt{2} x - 8 - 9 \sqrt{2})}{3} = \frac{0}{3}$

Step 5.2.2.4.1.2

Simplify the left side.

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Step 5.2.2.4.1.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.4.1.2.1.1

Cancel the common factor.

$\frac{3 (3 \sqrt{2} x - 8 - 9 \sqrt{2})}{3} = \frac{0}{3}$

Step 5.2.2.4.1.2.1.2

Divide $3 \sqrt{2} x - 8 - 9 \sqrt{2}$ by $1$ .

$3 \sqrt{2} x - 8 - 9 \sqrt{2} = \frac{0}{3}$

Step 5.2.2.4.1.3

Simplify the right side.

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Step 5.2.2.4.1.3.1

Divide $0$ by $3$ .

$3 \sqrt{2} x - 8 - 9 \sqrt{2} = 0$

Step 5.2.2.4.2

Move all terms not containing $x$ to the right side of the equation.

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Step 5.2.2.4.2.1

Add $8$ to both sides of the equation.

$3 \sqrt{2} x - 9 \sqrt{2} = 8$

Step 5.2.2.4.2.2

Add $9 \sqrt{2}$ to both sides of the equation.

$3 \sqrt{2} x = 8 + 9 \sqrt{2}$

Step 5.2.2.4.3

Divide each term in $3 \sqrt{2} x = 8 + 9 \sqrt{2}$ by $3 \sqrt{2}$ and simplify.

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Step 5.2.2.4.3.1

Divide each term in $3 \sqrt{2} x = 8 + 9 \sqrt{2}$ by $3 \sqrt{2}$ .

$\frac{3 \sqrt{2} x}{3 \sqrt{2}} = \frac{8}{3 \sqrt{2}} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.2

Simplify the left side.

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Step 5.2.2.4.3.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.4.3.2.1.1

Cancel the common factor.

$\frac{3 \sqrt{2} x}{3 \sqrt{2}} = \frac{8}{3 \sqrt{2}} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.2.1.2

Rewrite the expression.

$\frac{\sqrt{2} x}{\sqrt{2}} = \frac{8}{3 \sqrt{2}} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.2.2

Cancel the common factor of $\sqrt{2}$ .

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Step 5.2.2.4.3.2.2.1

Cancel the common factor.

$\frac{\sqrt{2} x}{\sqrt{2}} = \frac{8}{3 \sqrt{2}} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.2.2.2

Divide $x$ by $1$ .

$x = \frac{8}{3 \sqrt{2}} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3

Simplify the right side.

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Step 5.2.2.4.3.3.1

Simplify each term.

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Step 5.2.2.4.3.3.1.1

Multiply $\frac{8}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \frac{8}{3 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.2

Combine and simplify the denominator.

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Step 5.2.2.4.3.3.1.2.1

Multiply $\frac{8}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \frac{8 \sqrt{2}}{3 \sqrt{2} \sqrt{2}} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.2.2

Move $\sqrt{2}$ .

$x = \frac{8 \sqrt{2}}{3 (\sqrt{2} \sqrt{2})} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.2.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \frac{8 \sqrt{2}}{3 ({\sqrt{2}}^{1} \sqrt{2})} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.2.4

Raise $\sqrt{2}$ to the power of $1$ .

$x = \frac{8 \sqrt{2}}{3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.2.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{8 \sqrt{2}}{3 {\sqrt{2}}^{1 + 1}} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.2.6

Add $1$ and $1$ .

$x = \frac{8 \sqrt{2}}{3 {\sqrt{2}}^{2}} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.2.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 5.2.2.4.3.3.1.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \frac{8 \sqrt{2}}{3 {(2^{\frac{1}{2}})}^{2}} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.2.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{8 \sqrt{2}}{3 \cdot 2^{\frac{1}{2} \cdot 2}} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.2.7.3

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

$x = \frac{8 \sqrt{2}}{3 \cdot 2^{\frac{2}{2}}} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.2.7.4

Cancel the common factor of $2$ .

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Step 5.2.2.4.3.3.1.2.7.4.1

Cancel the common factor.

$x = \frac{8 \sqrt{2}}{3 \cdot 2^{\frac{2}{2}}} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.2.7.4.2

Rewrite the expression.

$x = \frac{8 \sqrt{2}}{3 \cdot 2^{1}} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.2.7.5

Evaluate the exponent.

$x = \frac{8 \sqrt{2}}{3 \cdot 2} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.3

Cancel the common factor of $8$ and $2$ .

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Step 5.2.2.4.3.3.1.3.1

Factor $2$ out of $8 \sqrt{2}$ .

$x = \frac{2 (4 \sqrt{2})}{3 \cdot 2} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.3.2

Cancel the common factors.

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Step 5.2.2.4.3.3.1.3.2.1

Factor $2$ out of $3 \cdot 2$ .

$x = \frac{2 (4 \sqrt{2})}{2 \cdot 3} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.3.2.2

Cancel the common factor.

$x = \frac{2 (4 \sqrt{2})}{2 \cdot 3} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.3.2.3

Rewrite the expression.

$x = \frac{4 \sqrt{2}}{3} + \frac{9 \sqrt{2}}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.4

Cancel the common factor of $9$ and $3$ .

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Step 5.2.2.4.3.3.1.4.1

Factor $3$ out of $9 \sqrt{2}$ .

$x = \frac{4 \sqrt{2}}{3} + \frac{3 (3 \sqrt{2})}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.4.2

Cancel the common factor.

$x = \frac{4 \sqrt{2}}{3} + \frac{3 (3 \sqrt{2})}{3 \sqrt{2}}$

Step 5.2.2.4.3.3.1.4.3

Rewrite the expression.

$x = \frac{4 \sqrt{2}}{3} + 3$

Step 5.2.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{4 \sqrt{2}}{3} + 3, 0)$

Step 5.3

Find the y-intercept.

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Step 5.3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{9 \sqrt{2}}{2} \cdot (0) - \frac{24 + 27 \sqrt{2}}{2}$

Step 5.3.2

Solve the equation.

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Step 5.3.2.1

Multiply $\frac{9 \sqrt{2}}{2}$ by $0$ .

$y = \frac{9 \sqrt{2}}{2} \cdot 0 - \frac{24 + 27 \sqrt{2}}{2}$

Step 5.3.2.2

Remove parentheses.

$y = \frac{9 \sqrt{2}}{2} \cdot (0) - \frac{24 + 27 \sqrt{2}}{2}$

Step 5.3.2.3

Simplify $\frac{9 \sqrt{2}}{2} \cdot (0) - \frac{24 + 27 \sqrt{2}}{2}$ .

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Step 5.3.2.3.1

Multiply $\frac{9 \sqrt{2}}{2}$ by $0$ .

$y = 0 - \frac{24 + 27 \sqrt{2}}{2}$

Step 5.3.2.3.2

Subtract $\frac{24 + 27 \sqrt{2}}{2}$ from $0$ .

$y = - \frac{24 + 27 \sqrt{2}}{2}$

Step 5.3.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{24 + 27 \sqrt{2}}{2})$

Step 5.4

Create a table of the $x$ and $y$ values.

$\begin{array}{c} x & y \\ 0 & - \frac{24 + 27 \sqrt{2}}{2} \\ \frac{4 \sqrt{2}}{3} + 3 & 0 \end{array}$

Step 6

Graph the line using the slope and the y-intercept, or the points.

Slope: $\frac{9}{2}$

y-intercept: $(0, - \frac{24 + 27 \sqrt{2}}{2})$

$\begin{array}{c} x & y \\ 0 & - \frac{24 + 27 \sqrt{2}}{2} \\ \frac{4 \sqrt{2}}{3} + 3 & 0 \end{array}$

Step 7