Precalculus Examples

$f (x) = - \frac{1}{2} \cos (3 t + 3 π) - 3$

Step 1

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

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

$b = 3$

$c = - 3 π$

$d = - 3$

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{2}$

Step 3

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

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Step 3.1

Find the period of $- \frac{\cos (3 t + 3 π)}{2}$ .

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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 $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 3.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Step 3.2

Find the period of $- 3$ .

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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 $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Step 3.3

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

$\frac{2 π}{3}$

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: $\frac{- 3 π}{3}$

Step 4.3

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

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Step 4.3.1

Factor $3$ out of $- 3 π$ .

Phase Shift: $\frac{3 (- π)}{3}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $3$ out of $3$ .

Phase Shift: $\frac{3 (- π)}{3 (1)}$

Step 4.3.2.2

Cancel the common factor.

Phase Shift: $\frac{3 (- π)}{3 \cdot 1}$

Step 4.3.2.3

Rewrite the expression.

Phase Shift: $\frac{- π}{1}$

Step 4.3.2.4

Divide $- π$ by $1$ .

Phase Shift: $- π$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{1}{2}$

Period: $\frac{2 π}{3}$

Phase Shift: $- π$ ( $π$ to the left)

Vertical Shift: $- 3$

Step 6