Algebra Examples

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

Step 1

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

$a = - 2$

$b = \frac{1}{2}$

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $2$

Step 3

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

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Step 3.1

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

$\frac{2 π}{| b |}$

Step 3.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 3.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 3.5

Multiply $2$ by $2$ .

$4 π$

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: $\frac{0}{\frac{1}{2}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 \cdot 2$

Step 4.4

Multiply $0$ by $2$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $2$

Period: $4 π$

Phase Shift: None

Vertical Shift: None

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = 0$ .

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Step 6.1.1

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

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

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Divide $0$ by $2$ .

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

Step 6.1.2.2

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

$f (0) = - 2 \cdot 1$

Step 6.1.2.3

Multiply $- 2$ by $1$ .

$f (0) = - 2$

Step 6.1.2.4

The final answer is $- 2$ .

$- 2$

Step 6.2

Find the point at $x = π$ .

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Step 6.2.1

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

$f (π) = - 2 \cos (\frac{π}{2})$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

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

$f (π) = - 2 \cdot 0$

Step 6.2.2.2

Multiply $- 2$ by $0$ .

$f (π) = 0$

Step 6.2.2.3

The final answer is $0$ .

$0$

Step 6.3

Find the point at $x = 2 π$ .

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Step 6.3.1

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

$f (2 π) = - 2 \cos (\frac{2 π}{2})$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Cancel the common factor of $2$ .

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Step 6.3.2.1.1

Cancel the common factor.

$f (2 π) = - 2 \cos (\frac{2 π}{2})$

Step 6.3.2.1.2

Divide $π$ by $1$ .

$f (2 π) = - 2 \cos (π)$

Step 6.3.2.2

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.

$f (2 π) = - 2 (- \cos (0))$

Step 6.3.2.3

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

$f (2 π) = - 2 (- 1 \cdot 1)$

Step 6.3.2.4

Multiply $- 2 (- 1 \cdot 1)$ .

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Step 6.3.2.4.1

Multiply $- 1$ by $1$ .

$f (2 π) = - 2 \cdot - 1$

Step 6.3.2.4.2

Multiply $- 2$ by $- 1$ .

$f (2 π) = 2$

Step 6.3.2.5

The final answer is $2$ .

$2$

Step 6.4

Find the point at $x = 3 π$ .

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Step 6.4.1

Replace the variable $x$ with $3 π$ in the expression.

$f (3 π) = - 2 \cos (\frac{3 π}{2})$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

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

$f (3 π) = - 2 \cos (\frac{π}{2})$

Step 6.4.2.2

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

$f (3 π) = - 2 \cdot 0$

Step 6.4.2.3

Multiply $- 2$ by $0$ .

$f (3 π) = 0$

Step 6.4.2.4

The final answer is $0$ .

$0$

Step 6.5

Find the point at $x = 4 π$ .

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Step 6.5.1

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

$f (4 π) = - 2 \cos (\frac{4 π}{2})$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

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

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Step 6.5.2.1.1

Factor $2$ out of $4 π$ .

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

Step 6.5.2.1.2

Cancel the common factors.

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Step 6.5.2.1.2.1

Factor $2$ out of $2$ .

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

Step 6.5.2.1.2.2

Cancel the common factor.

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

Step 6.5.2.1.2.3

Rewrite the expression.

$f (4 π) = - 2 \cos (\frac{2 π}{1})$

Step 6.5.2.1.2.4

Divide $2 π$ by $1$ .

$f (4 π) = - 2 \cos (2 π)$

Step 6.5.2.2

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

$f (4 π) = - 2 \cos (0)$

Step 6.5.2.3

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

$f (4 π) = - 2 \cdot 1$

Step 6.5.2.4

Multiply $- 2$ by $1$ .

$f (4 π) = - 2$

Step 6.5.2.5

The final answer is $- 2$ .

$- 2$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & - 2 \\ π & 0 \\ 2 π & 2 \\ 3 π & 0 \\ 4 π & - 2 \end{array}$

Step 7

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

Amplitude: $2$

Period: $4 π$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & - 2 \\ π & 0 \\ 2 π & 2 \\ 3 π & 0 \\ 4 π & - 2 \end{array}$

Step 8