Graph y=cos(2x+(3pi)/4)+1

$y = \cos (2 x + \frac{3 π}{4}) + 1$

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

$a = 1$

$b = 2$

$c = - \frac{3 π}{4}$

$d = 1$

Find the amplitude $| a |$ .

Amplitude: $1$

Find the period of $\cos (2 x + \frac{3 π}{4})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

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

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

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: $\frac{- \frac{3 π}{4}}{2}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{3 π}{4} \cdot \frac{1}{2}$

Multiply $- \frac{3 π}{4} \cdot \frac{1}{2}$ .

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Multiply $\frac{1}{2}$ and $\frac{3 π}{4}$ .

Phase Shift: $- \frac{3 π}{2 \cdot 4}$

Multiply $2$ by $4$ .

Phase Shift: $- \frac{3 π}{8}$

Find the vertical shift $d$ .

Vertical Shift: $1$

List the properties of the trigonometric function.

Amplitude: $1$

Period: $π$

Phase Shift: $- \frac{3 π}{8}$ ( $\frac{3 π}{8}$ to the left)

Vertical Shift: $1$

Select a few points to graph.

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Find the point at $x = - \frac{3 π}{8}$ .

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Replace the variable $x$ with $- \frac{3 π}{8}$ in the expression.

$f (- \frac{3 π}{8}) = \cos (2 (- \frac{3 π}{8}) + \frac{3 π}{4}) + 1$

Simplify the result.

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Simplify each term.

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Simplify each term.

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Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{3 π}{8}$ into the numerator.

$f (- \frac{3 π}{8}) = \cos (2 (\frac{- 3 π}{8}) + \frac{3 π}{4}) + 1$

Factor $2$ out of $8$ .

$f (- \frac{3 π}{8}) = \cos (2 (\frac{- 3 π}{2 (4)}) + \frac{3 π}{4}) + 1$

Cancel the common factor.

$f (- \frac{3 π}{8}) = \cos (2 (\frac{- 3 π}{2 \cdot 4}) + \frac{3 π}{4}) + 1$

Rewrite the expression.

$f (- \frac{3 π}{8}) = \cos (\frac{- 3 π}{4} + \frac{3 π}{4}) + 1$

Move the negative in front of the fraction.

$f (- \frac{3 π}{8}) = \cos (- \frac{3 π}{4} + \frac{3 π}{4}) + 1$

Combine the numerators over the common denominator.

$f (- \frac{3 π}{8}) = \cos (\frac{- 3 π + 3 π}{4}) + 1$

Add $- 3 π$ and $3 π$ .

$f (- \frac{3 π}{8}) = \cos (\frac{0}{4}) + 1$

Divide $0$ by $4$ .

$f (- \frac{3 π}{8}) = \cos (0) + 1$

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

$f (- \frac{3 π}{8}) = 1 + 1$

Add $1$ and $1$ .

$f (- \frac{3 π}{8}) = 2$

The final answer is $2$ .

$2$

Find the point at $x = - \frac{π}{8}$ .

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Replace the variable $x$ with $- \frac{π}{8}$ in the expression.

$f (- \frac{π}{8}) = \cos (2 (- \frac{π}{8}) + \frac{3 π}{4}) + 1$

Simplify the result.

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Simplify each term.

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Simplify each term.

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Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{π}{8}$ into the numerator.

$f (- \frac{π}{8}) = \cos (2 (\frac{- π}{8}) + \frac{3 π}{4}) + 1$

Factor $2$ out of $8$ .

$f (- \frac{π}{8}) = \cos (2 (\frac{- π}{2 (4)}) + \frac{3 π}{4}) + 1$

Cancel the common factor.

$f (- \frac{π}{8}) = \cos (2 (\frac{- π}{2 \cdot 4}) + \frac{3 π}{4}) + 1$

Rewrite the expression.

$f (- \frac{π}{8}) = \cos (\frac{- π}{4} + \frac{3 π}{4}) + 1$

Move the negative in front of the fraction.

$f (- \frac{π}{8}) = \cos (- \frac{π}{4} + \frac{3 π}{4}) + 1$

Combine the numerators over the common denominator.

$f (- \frac{π}{8}) = \cos (\frac{- π + 3 π}{4}) + 1$

Add $- π$ and $3 π$ .

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

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

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Factor $2$ out of $2 π$ .

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

Cancel the common factors.

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Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

$f (- \frac{π}{8}) = 0 + 1$

Add $0$ and $1$ .

$f (- \frac{π}{8}) = 1$

The final answer is $1$ .

$1$

Find the point at $x = \frac{π}{8}$ .

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Replace the variable $x$ with $\frac{π}{8}$ in the expression.

$f (\frac{π}{8}) = \cos (2 (\frac{π}{8}) + \frac{3 π}{4}) + 1$

Simplify the result.

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Simplify each term.

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Cancel the common factor of $2$ .

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Factor $2$ out of $8$ .

$f (\frac{π}{8}) = \cos (2 (\frac{π}{2 (4)}) + \frac{3 π}{4}) + 1$

Cancel the common factor.

$f (\frac{π}{8}) = \cos (2 (\frac{π}{2 \cdot 4}) + \frac{3 π}{4}) + 1$

Rewrite the expression.

$f (\frac{π}{8}) = \cos (\frac{π}{4} + \frac{3 π}{4}) + 1$

Combine the numerators over the common denominator.

$f (\frac{π}{8}) = \cos (\frac{π + 3 π}{4}) + 1$

Add $π$ and $3 π$ .

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

Cancel the common factor of $4$ .

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Cancel the common factor.

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

Divide $π$ by $1$ .

$f (\frac{π}{8}) = \cos (π) + 1$

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.

$f (\frac{π}{8}) = - \cos (0) + 1$

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

$f (\frac{π}{8}) = - 1 \cdot 1 + 1$

Multiply $- 1$ by $1$ .

$f (\frac{π}{8}) = - 1 + 1$

Add $- 1$ and $1$ .

$f (\frac{π}{8}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{3 π}{8}$ .

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Replace the variable $x$ with $\frac{3 π}{8}$ in the expression.

$f (\frac{3 π}{8}) = \cos (2 (\frac{3 π}{8}) + \frac{3 π}{4}) + 1$

Simplify the result.

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Simplify each term.

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Cancel the common factor of $2$ .

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Factor $2$ out of $8$ .

$f (\frac{3 π}{8}) = \cos (2 (\frac{3 π}{2 (4)}) + \frac{3 π}{4}) + 1$

Cancel the common factor.

$f (\frac{3 π}{8}) = \cos (2 (\frac{3 π}{2 \cdot 4}) + \frac{3 π}{4}) + 1$

Rewrite the expression.

$f (\frac{3 π}{8}) = \cos (\frac{3 π}{4} + \frac{3 π}{4}) + 1$

Combine the numerators over the common denominator.

$f (\frac{3 π}{8}) = \cos (\frac{3 π + 3 π}{4}) + 1$

Add $3 π$ and $3 π$ .

$f (\frac{3 π}{8}) = \cos (\frac{6 π}{4}) + 1$

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

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Factor $2$ out of $6 π$ .

$f (\frac{3 π}{8}) = \cos (\frac{2 (3 π)}{4}) + 1$

Cancel the common factors.

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Factor $2$ out of $4$ .

$f (\frac{3 π}{8}) = \cos (\frac{2 (3 π)}{2 (2)}) + 1$

Cancel the common factor.

$f (\frac{3 π}{8}) = \cos (\frac{2 (3 π)}{2 \cdot 2}) + 1$

Rewrite the expression.

$f (\frac{3 π}{8}) = \cos (\frac{3 π}{2}) + 1$

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

$f (\frac{3 π}{8}) = \cos (\frac{π}{2}) + 1$

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

$f (\frac{3 π}{8}) = 0 + 1$

Add $0$ and $1$ .

$f (\frac{3 π}{8}) = 1$

The final answer is $1$ .

$1$

Find the point at $x = \frac{5 π}{8}$ .

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Replace the variable $x$ with $\frac{5 π}{8}$ in the expression.

$f (\frac{5 π}{8}) = \cos (2 (\frac{5 π}{8}) + \frac{3 π}{4}) + 1$

Simplify the result.

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Simplify each term.

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Cancel the common factor of $2$ .

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Factor $2$ out of $8$ .

$f (\frac{5 π}{8}) = \cos (2 (\frac{5 π}{2 (4)}) + \frac{3 π}{4}) + 1$

Cancel the common factor.

$f (\frac{5 π}{8}) = \cos (2 (\frac{5 π}{2 \cdot 4}) + \frac{3 π}{4}) + 1$

Rewrite the expression.

$f (\frac{5 π}{8}) = \cos (\frac{5 π}{4} + \frac{3 π}{4}) + 1$

Combine the numerators over the common denominator.

$f (\frac{5 π}{8}) = \cos (\frac{5 π + 3 π}{4}) + 1$

Add $5 π$ and $3 π$ .

$f (\frac{5 π}{8}) = \cos (\frac{8 π}{4}) + 1$

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

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Factor $4$ out of $8 π$ .

$f (\frac{5 π}{8}) = \cos (\frac{4 (2 π)}{4}) + 1$

Cancel the common factors.

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Factor $4$ out of $4$ .

$f (\frac{5 π}{8}) = \cos (\frac{4 (2 π)}{4 (1)}) + 1$

Cancel the common factor.

$f (\frac{5 π}{8}) = \cos (\frac{4 (2 π)}{4 \cdot 1}) + 1$

Rewrite the expression.

$f (\frac{5 π}{8}) = \cos (\frac{2 π}{1}) + 1$

Divide $2 π$ by $1$ .

$f (\frac{5 π}{8}) = \cos (2 π) + 1$

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

$f (\frac{5 π}{8}) = \cos (0) + 1$

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

$f (\frac{5 π}{8}) = 1 + 1$

Add $1$ and $1$ .

$f (\frac{5 π}{8}) = 2$

The final answer is $2$ .

$2$

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{3 π}{8} & 2 \\ - \frac{π}{8} & 1 \\ \frac{π}{8} & 0 \\ \frac{3 π}{8} & 1 \\ \frac{5 π}{8} & 2 \end{array}$

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

Amplitude: $1$

Period: $π$

Phase Shift: $- \frac{3 π}{8}$ ( $\frac{3 π}{8}$ to the left)

Vertical Shift: $1$

$\begin{array}{c} x & f (x) \\ - \frac{3 π}{8} & 2 \\ - \frac{π}{8} & 1 \\ \frac{π}{8} & 0 \\ \frac{3 π}{8} & 1 \\ \frac{5 π}{8} & 2 \end{array}$

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