Precalculus Examples

$y = - 2 + 3 \cos (\frac{2}{3} x - \frac{π}{3})$

Step 1

Rewrite the expression as $3 \cos (\frac{2 x}{3} - \frac{π}{3}) - 2$ .

$3 \cos (\frac{2 x}{3} - \frac{π}{3}) - 2$

Step 2

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

$a = 3$

$b = \frac{2}{3}$

$c = \frac{π}{3}$

$d = - 2$

Step 3

Find the amplitude $| a |$ .

Amplitude: $3$

Step 4

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

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Step 4.1

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

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Step 4.1.1

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

$\frac{2 π}{| b |}$

Step 4.1.2

Replace $b$ with $\frac{2}{3}$ in the formula for period.

$\frac{2 π}{| \frac{2}{3} |}$

Step 4.1.3

$\frac{2}{3}$ is approximately $0.\overline{6}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2}{3}}$

Step 4.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{2}$

Step 4.1.5

Cancel the common factor of $2$ .

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Step 4.1.5.1

Factor $2$ out of $2 π$ .

$2 (π) \frac{3}{2}$

Step 4.1.5.2

Cancel the common factor.

$2 π \frac{3}{2}$

Step 4.1.5.3

Rewrite the expression.

$π \cdot 3$

Step 4.1.6

Move $3$ to the left of $π$ .

$3 π$

Step 4.2

Find the period of $- 2$ .

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Step 4.2.1

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

$\frac{2 π}{| b |}$

Step 4.2.2

Replace $b$ with $\frac{2}{3}$ in the formula for period.

$\frac{2 π}{| \frac{2}{3} |}$

Step 4.2.3

$\frac{2}{3}$ is approximately $0.\overline{6}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2}{3}}$

Step 4.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{2}$

Step 4.2.5

Cancel the common factor of $2$ .

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Step 4.2.5.1

Factor $2$ out of $2 π$ .

$2 (π) \frac{3}{2}$

Step 4.2.5.2

Cancel the common factor.

$2 π \frac{3}{2}$

Step 4.2.5.3

Rewrite the expression.

$π \cdot 3$

Step 4.2.6

Move $3$ to the left of $π$ .

$3 π$

Step 4.3

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

$3 π$

Step 5

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

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Step 5.1

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

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

Step 5.2

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

Phase Shift: $\frac{\frac{π}{3}}{\frac{2}{3}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{3} \cdot \frac{3}{2}$

Step 5.4

Cancel the common factor of $3$ .

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Step 5.4.1

Cancel the common factor.

Phase Shift: $\frac{π}{3} \cdot \frac{3}{2}$

Step 5.4.2

Rewrite the expression.

Phase Shift: $π (\frac{1}{2})$

Step 5.5

Combine $π$ and $\frac{1}{2}$ .

Phase Shift: $\frac{π}{2}$

Step 6

List the properties of the trigonometric function.

Amplitude: $3$

Period: $3 π$

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

Vertical Shift: $- 2$

Step 7