Trigonometry Examples

$y = 240 \sin (100 π x)$

Step 1

Use the form $a \sin (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = 240$

$b = 100 π$

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $240$

Step 3

Find the period of $240 \sin (100 π x)$ .

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Step 3.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2

Replace $b$ with $100 π$ in the formula for period.

$\frac{2 π}{| 100 π |}$

Step 3.3

$100 π$ is approximately $314.15926535$ which is positive so remove the absolute value

$\frac{2 π}{100 π}$

Step 3.4

Cancel the common factor of $2$ and $100$ .

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Step 3.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{100 π}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $100 π$ .

$\frac{2 (π)}{2 (50 π)}$

Step 3.4.2.2

Cancel the common factor.

$\frac{2 π}{2 (50 π)}$

Step 3.4.2.3

Rewrite the expression.

$\frac{π}{50 π}$

Step 3.5

Cancel the common factor of $π$ .

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Step 3.5.1

Cancel the common factor.

$\frac{π}{50 π}$

Step 3.5.2

Rewrite the expression.

$\frac{1}{50}$

Step 4

Find the phase shift using the formula $\frac{c}{b}$ .

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Step 4.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 4.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{0}{100 π}$

Step 4.3

Cancel the common factor of $0$ and $100$ .

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Step 4.3.1

Factor $100$ out of $0$ .

Phase Shift: $\frac{100 (0)}{100 π}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $100$ out of $100 π$ .

Phase Shift: $\frac{100 (0)}{100 (π)}$

Step 4.3.2.2

Cancel the common factor.

Phase Shift: $\frac{100 \cdot 0}{100 π}$

Step 4.3.2.3

Rewrite the expression.

Phase Shift: $\frac{0}{π}$

Step 4.4

Divide $0$ by $π$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $240$

Period: $\frac{1}{50}$

Phase Shift: None

Vertical Shift: None

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = 0$ .

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Step 6.1.1

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

$f (0) = 240 \sin (100 π (0))$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Multiply $100 π (0)$ .

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Step 6.1.2.1.1

Multiply $0$ by $100$ .

$f (0) = 240 \sin (0 π)$

Step 6.1.2.1.2

Multiply $0$ by $π$ .

$f (0) = 240 \sin (0)$

Step 6.1.2.2

The exact value of $\sin (0)$ is $0$ .

$f (0) = 240 \cdot 0$

Step 6.1.2.3

Multiply $240$ by $0$ .

$f (0) = 0$

Step 6.1.2.4

The final answer is $0$ .

$0$

Step 6.2

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ \frac{1}{100} & 0 \\ \frac{1}{50} & 0 \\ \frac{3}{100} & 0 \\ \frac{1}{25} & 0 \end{array}$

Step 7

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

Amplitude: $240$

Period: $\frac{1}{50}$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ \frac{1}{100} & 0 \\ \frac{1}{50} & 0 \\ \frac{3}{100} & 0 \\ \frac{1}{25} & 0 \end{array}$

Step 8