Precalculus Examples

$y = 1 + \cos (2 x + \frac{π}{3})$

Step 1

Rewrite the expression as $\cos (2 x + \frac{π}{3}) + 1$ .

$\cos (2 x + \frac{π}{3}) + 1$

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.

$a = 1$

$b = 2$

$c = - \frac{π}{3}$

$d = 1$

Step 3

Find the amplitude $| a |$ .

Amplitude: $1$

Step 4

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

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Step 4.1

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

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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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 4.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 4.1.4.2

Divide $π$ by $1$ .

$π$

Step 4.2

Find the period of $1$ .

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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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 4.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 4.2.4.2

Divide $π$ by $1$ .

$π$

Step 4.3

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

$π$

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: $\frac{- \frac{π}{3}}{2}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{3} \cdot \frac{1}{2}$

Step 5.4

Multiply $- \frac{π}{3} \cdot \frac{1}{2}$ .

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Step 5.4.1

Multiply $\frac{1}{2}$ by $\frac{π}{3}$ .

Phase Shift: $- \frac{π}{2 \cdot 3}$

Step 5.4.2

Multiply $2$ by $3$ .

Phase Shift: $- \frac{π}{6}$

Step 6

List the properties of the trigonometric function.

Amplitude: $1$

Period: $π$

Phase Shift: $- \frac{π}{6}$ ( $\frac{π}{6}$ to the left)

Vertical Shift: $1$

Step 7