Trigonometry Examples

$y = - \sin (- π x + \frac{π}{2}) - 2$

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.

$a = - 1$

$b = - π$

$c = - \frac{π}{2}$

$d = - 2$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

Find the period using the formula $\frac{2 π}{| b |}$ .

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Step 3.1

Find the period of $- \sin (- π x + \frac{π}{2})$ .

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Step 3.1.1

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

$\frac{2 π}{| b |}$

Step 3.1.2

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

$\frac{2 π}{| - π |}$

Step 3.1.3

$- π$ is approximately $- 3.14159265$ which is negative so negate $- π$ and remove the absolute value

$\frac{2 π}{π}$

Step 3.1.4

Cancel the common factor of $π$ .

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Step 3.1.4.1

Cancel the common factor.

$\frac{2 π}{π}$

Step 3.1.4.2

Divide $2$ by $1$ .

$2$

Step 3.2

Find the period of $- 2$ .

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Step 3.2.1

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

$\frac{2 π}{| b |}$

Step 3.2.2

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

$\frac{2 π}{| - π |}$

Step 3.2.3

$- π$ is approximately $- 3.14159265$ which is negative so negate $- π$ and remove the absolute value

$\frac{2 π}{π}$

Step 3.2.4

Cancel the common factor of $π$ .

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Step 3.2.4.1

Cancel the common factor.

$\frac{2 π}{π}$

Step 3.2.4.2

Divide $2$ by $1$ .

$2$

Step 3.3

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$2$

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: $\frac{- \frac{π}{2}}{- π}$

Step 4.3

Dividing two negative values results in a positive value.

Phase Shift: $\frac{\frac{π}{2}}{π}$

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{2} \cdot \frac{1}{π}$

Step 4.5

Cancel the common factor of $π$ .

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Step 4.5.1

Cancel the common factor.

Phase Shift: $\frac{π}{2} \cdot \frac{1}{π}$

Step 4.5.2

Rewrite the expression.

Phase Shift: $\frac{1}{2}$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $2$

Phase Shift: $\frac{1}{2}$ ( $\frac{1}{2}$ to the right)

Vertical Shift: $- 2$

Step 6