Algebra Examples

$f (x) = \cos (x)$ $g (x) = \cos (x + \frac{π}{2})$

Step 1

Graph $f (x) = \cos (x)$ .

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Step 1.1

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

$a = 1$

$b = 1$

$c = 0$

$d = 0$

Step 1.2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 1.3

Find the period of $\cos (x)$ .

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Step 1.3.1

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

$\frac{2 π}{| b |}$

Step 1.3.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 1.3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 1.3.4

Divide $2 π$ by $1$ .

$2 π$

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Divide $0$ by $1$ .

Phase Shift: $0$

Step 1.5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $2 π$

Phase Shift: None

Vertical Shift: None

Step 1.6

Select a few points to graph.

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Step 1.6.1

Find the point at $x = 0$ .

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Step 1.6.1.1

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

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

Step 1.6.1.2

Simplify the result.

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Step 1.6.1.2.1

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

$f (0) = 1$

Step 1.6.1.2.2

The final answer is $1$ .

$1$

Step 1.6.2

Find the point at $x = \frac{π}{2}$ .

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Step 1.6.2.1

Replace the variable $x$ with $\frac{π}{2}$ in the expression.

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

Step 1.6.2.2

Simplify the result.

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Step 1.6.2.2.1

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

$f (\frac{π}{2}) = 0$

Step 1.6.2.2.2

The final answer is $0$ .

$0$

Step 1.6.3

Find the point at $x = π$ .

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Step 1.6.3.1

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

$f (π) = \cos (π)$

Step 1.6.3.2

Simplify the result.

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Step 1.6.3.2.1

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.

$f (π) = - \cos (0)$

Step 1.6.3.2.2

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

$f (π) = - 1 \cdot 1$

Step 1.6.3.2.3

Multiply $- 1$ by $1$ .

$f (π) = - 1$

Step 1.6.3.2.4

The final answer is $- 1$ .

$- 1$

Step 1.6.4

Find the point at $x = \frac{3 π}{2}$ .

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Step 1.6.4.1

Replace the variable $x$ with $\frac{3 π}{2}$ in the expression.

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

Step 1.6.4.2

Simplify the result.

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Step 1.6.4.2.1

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

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

Step 1.6.4.2.2

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

$f (\frac{3 π}{2}) = 0$

Step 1.6.4.2.3

The final answer is $0$ .

$0$

Step 1.6.5

Find the point at $x = 2 π$ .

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Step 1.6.5.1

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

$f (2 π) = \cos (2 π)$

Step 1.6.5.2

Simplify the result.

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Step 1.6.5.2.1

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

$f (2 π) = \cos (0)$

Step 1.6.5.2.2

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

$f (2 π) = 1$

Step 1.6.5.2.3

The final answer is $1$ .

$1$

Step 1.6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 1 \\ \frac{π}{2} & 0 \\ π & - 1 \\ \frac{3 π}{2} & 0 \\ 2 π & 1 \end{array}$

Step 1.7

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

Amplitude: $1$

Period: $2 π$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & 1 \\ \frac{π}{2} & 0 \\ π & - 1 \\ \frac{3 π}{2} & 0 \\ 2 π & 1 \end{array}$

Amplitude: $1$

Period: $2 π$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & 1 \\ \frac{π}{2} & 0 \\ π & - 1 \\ \frac{3 π}{2} & 0 \\ 2 π & 1 \end{array}$

Step 2

Graph $g (x) = \cos (x + \frac{π}{2})$ .

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Step 2.1

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

$a = 1$

$b = 1$

$c = - \frac{π}{2}$

$d = 0$

Step 2.2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 2.3

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

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Step 2.3.1

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

$\frac{2 π}{| b |}$

Step 2.3.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 2.3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 2.3.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

Divide $- \frac{π}{2}$ by $1$ .

Phase Shift: $- \frac{π}{2}$

Step 2.5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $2 π$

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

Vertical Shift: None

Step 2.6

Select a few points to graph.

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Step 2.6.1

Find the point at $x = - \frac{π}{2}$ .

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Step 2.6.1.1

Replace the variable $x$ with $- \frac{π}{2}$ in the expression.

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

Step 2.6.1.2

Simplify the result.

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Step 2.6.1.2.1

Combine the numerators over the common denominator.

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

Step 2.6.1.2.2

Add $- π$ and $π$ .

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

Step 2.6.1.2.3

Divide $0$ by $2$ .

$f (- \frac{π}{2}) = \cos (0)$

Step 2.6.1.2.4

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

$f (- \frac{π}{2}) = 1$

Step 2.6.1.2.5

The final answer is $1$ .

$1$

Step 2.6.2

Find the point at $x = 0$ .

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Step 2.6.2.1

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

$f (0) = \cos ((0) + \frac{π}{2})$

Step 2.6.2.2

Simplify the result.

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Step 2.6.2.2.1

Add $0$ and $\frac{π}{2}$ .

$f (0) = \cos (\frac{π}{2})$

Step 2.6.2.2.2

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

$f (0) = 0$

Step 2.6.2.2.3

The final answer is $0$ .

$0$

Step 2.6.3

Find the point at $x = \frac{π}{2}$ .

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Step 2.6.3.1

Replace the variable $x$ with $\frac{π}{2}$ in the expression.

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

Step 2.6.3.2

Simplify the result.

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Step 2.6.3.2.1

Combine the numerators over the common denominator.

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

Step 2.6.3.2.2

Add $π$ and $π$ .

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

Step 2.6.3.2.3

Cancel the common factor of $2$ .

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Step 2.6.3.2.3.1

Cancel the common factor.

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

Step 2.6.3.2.3.2

Divide $π$ by $1$ .

$f (\frac{π}{2}) = \cos (π)$

Step 2.6.3.2.4

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.

$f (\frac{π}{2}) = - \cos (0)$

Step 2.6.3.2.5

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

$f (\frac{π}{2}) = - 1 \cdot 1$

Step 2.6.3.2.6

Multiply $- 1$ by $1$ .

$f (\frac{π}{2}) = - 1$

Step 2.6.3.2.7

The final answer is $- 1$ .

$- 1$

Step 2.6.4

Find the point at $x = π$ .

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Step 2.6.4.1

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

$f (π) = \cos ((π) + \frac{π}{2})$

Step 2.6.4.2

Simplify the result.

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Step 2.6.4.2.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.6.4.2.2

Combine fractions.

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Step 2.6.4.2.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 2.6.4.2.2.2

Combine the numerators over the common denominator.

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

Step 2.6.4.2.3

Simplify the numerator.

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Step 2.6.4.2.3.1

Move $2$ to the left of $π$ .

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

Step 2.6.4.2.3.2

Add $2 π$ and $π$ .

$f (π) = \cos (\frac{3 π}{2})$

Step 2.6.4.2.4

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

$f (π) = \cos (\frac{π}{2})$

Step 2.6.4.2.5

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

$f (π) = 0$

Step 2.6.4.2.6

The final answer is $0$ .

$0$

Step 2.6.5

Find the point at $x = \frac{3 π}{2}$ .

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Step 2.6.5.1

Replace the variable $x$ with $\frac{3 π}{2}$ in the expression.

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

Step 2.6.5.2

Simplify the result.

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Step 2.6.5.2.1

Combine the numerators over the common denominator.

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

Step 2.6.5.2.2

Add $3 π$ and $π$ .

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

Step 2.6.5.2.3

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

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Step 2.6.5.2.3.1

Factor $2$ out of $4 π$ .

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

Step 2.6.5.2.3.2

Cancel the common factors.

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Step 2.6.5.2.3.2.1

Factor $2$ out of $2$ .

$f (\frac{3 π}{2}) = \cos (\frac{2 (2 π)}{2 (1)})$

Step 2.6.5.2.3.2.2

Cancel the common factor.

$f (\frac{3 π}{2}) = \cos (\frac{2 (2 π)}{2 \cdot 1})$

Step 2.6.5.2.3.2.3

Rewrite the expression.

$f (\frac{3 π}{2}) = \cos (\frac{2 π}{1})$

Step 2.6.5.2.3.2.4

Divide $2 π$ by $1$ .

$f (\frac{3 π}{2}) = \cos (2 π)$

Step 2.6.5.2.4

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

$f (\frac{3 π}{2}) = \cos (0)$

Step 2.6.5.2.5

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

$f (\frac{3 π}{2}) = 1$

Step 2.6.5.2.6

The final answer is $1$ .

$1$

Step 2.6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{π}{2} & 1 \\ 0 & 0 \\ \frac{π}{2} & - 1 \\ π & 0 \\ \frac{3 π}{2} & 1 \end{array}$

Step 2.7

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

Amplitude: $1$

Period: $2 π$

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

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - \frac{π}{2} & 1 \\ 0 & 0 \\ \frac{π}{2} & - 1 \\ π & 0 \\ \frac{3 π}{2} & 1 \end{array}$

Amplitude: $1$

Period: $2 π$

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

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - \frac{π}{2} & 1 \\ 0 & 0 \\ \frac{π}{2} & - 1 \\ π & 0 \\ \frac{3 π}{2} & 1 \end{array}$

Step 3

Plot each graph on the same coordinate system.

$f (x) = \cos (x)$

$g (x) = \cos (x + \frac{π}{2})$

Step 4