Graph y=-3/2-cos(x)

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

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

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

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

$a = - 1$

$b = 1$

$c = 0$

$d = - \frac{3}{2}$

Find the amplitude $| a |$ .

Amplitude: $1$

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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: $\frac{0}{1}$

Divide $0$ by $1$ .

Phase Shift: $0$

Find the vertical shift $d$ .

Vertical Shift: $- \frac{3}{2}$

List the properties of the trigonometric function.

Amplitude: $1$

Period: $2 π$

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

Vertical Shift: $- \frac{3}{2}$

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.

$f (0) = - \frac{3}{2} - \cos (0)$

Simplify the result.

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Simplify each term.

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

$f (0) = - \frac{3}{2} - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (0) = - \frac{3}{2} - 1$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (0) = - \frac{3}{2} - 1 \cdot \frac{2}{2}$

Combine $- 1$ and $\frac{2}{2}$ .

$f (0) = - \frac{3}{2} + \frac{- 1 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (0) = \frac{- 3 - 1 \cdot 2}{2}$

Simplify the numerator.

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Multiply $- 1$ by $2$ .

$f (0) = \frac{- 3 - 2}{2}$

Subtract $2$ from $- 3$ .

$f (0) = \frac{- 5}{2}$

Move the negative in front of the fraction.

$f (0) = - \frac{5}{2}$

The final answer is $- \frac{5}{2}$ .

$- \frac{5}{2}$

Find the point at $x = \frac{π}{2}$ .

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

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

Simplify the result.

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Simplify each term.

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

$f (\frac{π}{2}) = - \frac{3}{2} - 0$

Multiply $- 1$ by $0$ .

$f (\frac{π}{2}) = - \frac{3}{2} + 0$

Add $- \frac{3}{2}$ and $0$ .

$f (\frac{π}{2}) = - \frac{3}{2}$

The final answer is $- \frac{3}{2}$ .

$- \frac{3}{2}$

Find the point at $x = π$ .

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

$f (π) = - \frac{3}{2} - \cos (π)$

Simplify the result.

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Simplify each term.

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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.

$f (π) = - \frac{3}{2} + \cos (0)$

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

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

Multiply $- 1$ by $1$ .

$f (π) = - \frac{3}{2} + 1$

Multiply $- 1$ by $- 1$ .

$f (π) = - \frac{3}{2} + 1$

Simplify the expression.

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Write $1$ as a fraction with a common denominator.

$f (π) = - \frac{3}{2} + \frac{2}{2}$

Combine the numerators over the common denominator.

$f (π) = \frac{- 3 + 2}{2}$

Add $- 3$ and $2$ .

$f (π) = \frac{- 1}{2}$

Move the negative in front of the fraction.

$f (π) = - \frac{1}{2}$

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

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

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

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

Simplify the result.

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Simplify each term.

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

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

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

$f (\frac{3 π}{2}) = - \frac{3}{2} - 0$

Multiply $- 1$ by $0$ .

$f (\frac{3 π}{2}) = - \frac{3}{2} + 0$

Add $- \frac{3}{2}$ and $0$ .

$f (\frac{3 π}{2}) = - \frac{3}{2}$

The final answer is $- \frac{3}{2}$ .

$- \frac{3}{2}$

Find the point at $x = 2 π$ .

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

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

Simplify the result.

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Simplify each term.

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

$f (2 π) = - \frac{3}{2} - \cos (0)$

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

$f (2 π) = - \frac{3}{2} - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (2 π) = - \frac{3}{2} - 1$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine $- 1$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$f (2 π) = \frac{- 3 - 1 \cdot 2}{2}$

Simplify the numerator.

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Multiply $- 1$ by $2$ .

$f (2 π) = \frac{- 3 - 2}{2}$

Subtract $2$ from $- 3$ .

$f (2 π) = \frac{- 5}{2}$

Move the negative in front of the fraction.

$f (2 π) = - \frac{5}{2}$

The final answer is $- \frac{5}{2}$ .

$- \frac{5}{2}$

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & - \frac{5}{2} \\ \frac{π}{2} & - \frac{3}{2} \\ π & - \frac{1}{2} \\ \frac{3 π}{2} & - \frac{3}{2} \\ 2 π & - \frac{5}{2} \end{array}$

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

Amplitude: $1$

Period: $2 π$

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

Vertical Shift: $- \frac{3}{2}$

$\begin{array}{c} x & f (x) \\ 0 & - \frac{5}{2} \\ \frac{π}{2} & - \frac{3}{2} \\ π & - \frac{1}{2} \\ \frac{3 π}{2} & - \frac{3}{2} \\ 2 π & - \frac{5}{2} \end{array}$

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