Graph y=2cos(x/4)

$y = 2 \cos (\frac{x}{4})$

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

$a = 2$

$b = \frac{1}{4}$

$c = 0$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $2$

Find the period of $2 \cos (\frac{x}{4})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{4}$ in the formula for period.

$\frac{2 π}{| \frac{1}{4} |}$

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 4$

Multiply $4$ by $2$ .

$8 π$

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

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

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

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

Phase Shift: $\frac{0}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 \cdot 4$

Multiply $0$ by $4$ .

Phase Shift: $0$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: $2$

Period: $8 π$

Phase Shift: $0$ ( $0$ to the right)

Vertical Shift: $0$

Select a few points to graph.

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Find the point at $x = 0$ .

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

$f (0) = 2 \cos (\frac{0}{4})$

Simplify the result.

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Divide $0$ by $4$ .

$f (0) = 2 \cos (0)$

The exact value of $\cos (0)$ is $1$ .

$f (0) = 2 \cdot 1$

Multiply $2$ by $1$ .

$f (0) = 2$

The final answer is $2$ .

$2$

Find the point at $x = 2 π$ .

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Replace the variable $x$ with $2 π$ in the expression.

$f (2 π) = 2 \cos (\frac{2 π}{4})$

Simplify the result.

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Cancel the common factor of $2$ and $4$ .

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Factor $2$ out of $2 π$ .

$f (2 π) = 2 \cos (\frac{2 (π)}{4})$

Cancel the common factors.

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Factor $2$ out of $4$ .

$f (2 π) = 2 \cos (\frac{2 π}{2 \cdot 2})$

Cancel the common factor.

$f (2 π) = 2 \cos (\frac{2 π}{2 \cdot 2})$

Rewrite the expression.

$f (2 π) = 2 \cos (\frac{π}{2})$

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (2 π) = 2 \cdot 0$

Multiply $2$ by $0$ .

$f (2 π) = 0$

The final answer is $0$ .

$0$

Find the point at $x = 4 π$ .

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Replace the variable $x$ with $4 π$ in the expression.

$f (4 π) = 2 \cos (\frac{4 π}{4})$

Simplify the result.

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Cancel the common factor of $4$ .

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Cancel the common factor.

$f (4 π) = 2 \cos (\frac{4 π}{4})$

Divide $π$ by $1$ .

$f (4 π) = 2 \cos (π)$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$f (4 π) = 2 (- \cos (0))$

The exact value of $\cos (0)$ is $1$ .

$f (4 π) = 2 (- 1 \cdot 1)$

Multiply $- 1$ by $1$ .

$f (4 π) = 2 \cdot - 1$

Multiply $2$ by $- 1$ .

$f (4 π) = - 2$

The final answer is $- 2$ .

$- 2$

Find the point at $x = 6 π$ .

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Replace the variable $x$ with $6 π$ in the expression.

$f (6 π) = 2 \cos (\frac{6 π}{4})$

Simplify the result.

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Cancel the common factor of $6$ and $4$ .

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Factor $2$ out of $6 π$ .

$f (6 π) = 2 \cos (\frac{2 (3 π)}{4})$

Cancel the common factors.

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Factor $2$ out of $4$ .

$f (6 π) = 2 \cos (\frac{2 (3 π)}{2 (2)})$

Cancel the common factor.

$f (6 π) = 2 \cos (\frac{2 (3 π)}{2 \cdot 2})$

Rewrite the expression.

$f (6 π) = 2 \cos (\frac{3 π}{2})$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$f (6 π) = 2 \cos (\frac{π}{2})$

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (6 π) = 2 \cdot 0$

Multiply $2$ by $0$ .

$f (6 π) = 0$

The final answer is $0$ .

$0$

Find the point at $x = 8 π$ .

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Replace the variable $x$ with $8 π$ in the expression.

$f (8 π) = 2 \cos (\frac{8 π}{4})$

Simplify the result.

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Cancel the common factor of $8$ and $4$ .

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Factor $4$ out of $8 π$ .

$f (8 π) = 2 \cos (\frac{4 (2 π)}{4})$

Cancel the common factors.

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Factor $4$ out of $4$ .

$f (8 π) = 2 \cos (\frac{4 (2 π)}{4 (1)})$

Cancel the common factor.

$f (8 π) = 2 \cos (\frac{4 (2 π)}{4 \cdot 1})$

Rewrite the expression.

$f (8 π) = 2 \cos (\frac{2 π}{1})$

Divide $2 π$ by $1$ .

$f (8 π) = 2 \cos (2 π)$

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (8 π) = 2 \cos (0)$

The exact value of $\cos (0)$ is $1$ .

$f (8 π) = 2 \cdot 1$

Multiply $2$ by $1$ .

$f (8 π) = 2$

The final answer is $2$ .

$2$

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 2 \\ 2 π & 0 \\ 4 π & - 2 \\ 6 π & 0 \\ 8 π & 2 \end{array}$

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

Amplitude: $2$

Period: $8 π$

Phase Shift: $0$ ( $0$ to the right)

Vertical Shift: $0$

$\begin{array}{c} x & f (x) \\ 0 & 2 \\ 2 π & 0 \\ 4 π & - 2 \\ 6 π & 0 \\ 8 π & 2 \end{array}$

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