Trigonometry Examples

$f (x) = 8 (x + \frac{π}{2}) + 3$

Step 1

Rewrite the function as an equation.

$y = 8 x + 4 π + 3$

Step 2

The slope and y-intercept cannot be found for this problem since it is not linear.

Not Linear

Step 3

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

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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

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

$m =$

$b =$

Step 3.3

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

Slope:

y-intercept: $(0,)$

Slope:

y-intercept: $(0,)$

Step 4

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 4.1

Find the x-intercept.

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Step 4.1.1

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

$0 = 8 x + 4 π + 3$

Step 4.1.2

Solve the equation.

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Step 4.1.2.1

Rewrite the equation as $8 x + 4 π + 3 = 0$ .

$8 x + 4 π + 3 = 0$

Step 4.1.2.2

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

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Step 4.1.2.2.1

Subtract $4 π$ from both sides of the equation.

$8 x + 3 = - 4 π$

Step 4.1.2.2.2

Subtract $3$ from both sides of the equation.

$8 x = - 4 π - 3$

Step 4.1.2.3

Divide each term in $8 x = - 4 π - 3$ by $8$ and simplify.

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Step 4.1.2.3.1

Divide each term in $8 x = - 4 π - 3$ by $8$ .

$\frac{8 x}{8} = \frac{- 4 π}{8} + \frac{- 3}{8}$

Step 4.1.2.3.2

Simplify the left side.

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Step 4.1.2.3.2.1

Cancel the common factor of $8$ .

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Step 4.1.2.3.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 4 π}{8} + \frac{- 3}{8}$

Step 4.1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4 π}{8} + \frac{- 3}{8}$

Step 4.1.2.3.3

Simplify the right side.

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Step 4.1.2.3.3.1

Simplify each term.

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Step 4.1.2.3.3.1.1

Cancel the common factor of $- 4$ and $8$ .

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Step 4.1.2.3.3.1.1.1

Factor $4$ out of $- 4 π$ .

$x = \frac{4 (- π)}{8} + \frac{- 3}{8}$

Step 4.1.2.3.3.1.1.2

Cancel the common factors.

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Step 4.1.2.3.3.1.1.2.1

Factor $4$ out of $8$ .

$x = \frac{4 (- π)}{4 (2)} + \frac{- 3}{8}$

Step 4.1.2.3.3.1.1.2.2

Cancel the common factor.

$x = \frac{4 (- π)}{4 \cdot 2} + \frac{- 3}{8}$

Step 4.1.2.3.3.1.1.2.3

Rewrite the expression.

$x = \frac{- π}{2} + \frac{- 3}{8}$

Step 4.1.2.3.3.1.2

Move the negative in front of the fraction.

$x = - \frac{π}{2} + \frac{- 3}{8}$

Step 4.1.2.3.3.1.3

Move the negative in front of the fraction.

$x = - \frac{π}{2} - \frac{3}{8}$

Step 4.1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{π}{2} - \frac{3}{8}, 0)$

Step 4.2

Find the y-intercept.

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Step 4.2.1

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

$y = 8 (0) + 4 π + 3$

Step 4.2.2

Solve the equation.

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Step 4.2.2.1

Remove parentheses.

$y = 8 (0) + 4 π + 3$

Step 4.2.2.2

Simplify $8 (0) + 4 π + 3$ .

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Step 4.2.2.2.1

Multiply $8$ by $0$ .

$y = 0 + 4 π + 3$

Step 4.2.2.2.2

Add $0$ and $4 π$ .

$y = 4 π + 3$

Step 4.2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 4 π + 3)$

Step 4.3

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

$\begin{array}{c} x & y \\ - \frac{π}{2} - \frac{3}{8} & 0 \\ 0 & 4 π + 3 \end{array}$

Step 5

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

Slope:

y-intercept: $(0,)$

$\begin{array}{c} x & y \\ - \frac{π}{2} - \frac{3}{8} & 0 \\ 0 & 4 π + 3 \end{array}$

Step 6