Graph y=8-cos(3x+pi/4)

$y = 8 - \cos (3 x + \frac{π}{4})$

Rewrite the expression as $- \cos (3 x + \frac{π}{4}) + 8$ .

$- \cos (3 x + \frac{π}{4}) + 8$

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

$a = - 1$

$b = 3$

$c = - \frac{π}{4}$

$d = 8$

Find the amplitude $| a |$ .

Amplitude: $1$

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

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: $\frac{- \frac{π}{4}}{3}$

Multiply the numerator by the reciprocal of the denominator.

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

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

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

Phase Shift: $- \frac{π}{3 \cdot 4}$

Multiply $3$ by $4$ .

Phase Shift: $- \frac{π}{12}$

Find the vertical shift $d$ .

Vertical Shift: $8$

List the properties of the trigonometric function.

Amplitude: $1$

Period: $\frac{2 π}{3}$

Phase Shift: $- \frac{π}{12}$ ( $\frac{π}{12}$ to the left)

Vertical Shift: $8$

Select a few points to graph.

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Find the point at $x = - \frac{π}{12}$ .

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Replace the variable $x$ with $- \frac{π}{12}$ in the expression.

$f (- \frac{π}{12}) = 8 - \cos (3 (- \frac{π}{12}) + \frac{π}{4})$

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 $3$ .

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Move the leading negative in $- \frac{π}{12}$ into the numerator.

$f (- \frac{π}{12}) = 8 - \cos (3 (\frac{- π}{12}) + \frac{π}{4})$

Factor $3$ out of $12$ .

$f (- \frac{π}{12}) = 8 - \cos (3 (\frac{- π}{3 (4)}) + \frac{π}{4})$

Cancel the common factor.

$f (- \frac{π}{12}) = 8 - \cos (3 (\frac{- π}{3 \cdot 4}) + \frac{π}{4})$

Rewrite the expression.

$f (- \frac{π}{12}) = 8 - \cos (\frac{- π}{4} + \frac{π}{4})$

Move the negative in front of the fraction.

$f (- \frac{π}{12}) = 8 - \cos (- \frac{π}{4} + \frac{π}{4})$

Combine the numerators over the common denominator.

$f (- \frac{π}{12}) = 8 - \cos (\frac{- π + π}{4})$

Add $- π$ and $π$ .

$f (- \frac{π}{12}) = 8 - \cos (\frac{0}{4})$

Divide $0$ by $4$ .

$f (- \frac{π}{12}) = 8 - \cos (0)$

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

$f (- \frac{π}{12}) = 8 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (- \frac{π}{12}) = 8 - 1$

Subtract $1$ from $8$ .

$f (- \frac{π}{12}) = 7$

The final answer is $7$ .

$7$

Find the point at $x = \frac{π}{12}$ .

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Replace the variable $x$ with $\frac{π}{12}$ in the expression.

$f (\frac{π}{12}) = 8 - \cos (3 (\frac{π}{12}) + \frac{π}{4})$

Simplify the result.

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Simplify each term.

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Cancel the common factor of $3$ .

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Factor $3$ out of $12$ .

$f (\frac{π}{12}) = 8 - \cos (3 (\frac{π}{3 (4)}) + \frac{π}{4})$

Cancel the common factor.

$f (\frac{π}{12}) = 8 - \cos (3 (\frac{π}{3 \cdot 4}) + \frac{π}{4})$

Rewrite the expression.

$f (\frac{π}{12}) = 8 - \cos (\frac{π}{4} + \frac{π}{4})$

Combine the numerators over the common denominator.

$f (\frac{π}{12}) = 8 - \cos (\frac{π + π}{4})$

Add $π$ and $π$ .

$f (\frac{π}{12}) = 8 - \cos (\frac{2 π}{4})$

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

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Factor $2$ out of $2 π$ .

$f (\frac{π}{12}) = 8 - \cos (\frac{2 (π)}{4})$

Cancel the common factors.

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Factor $2$ out of $4$ .

$f (\frac{π}{12}) = 8 - \cos (\frac{2 π}{2 \cdot 2})$

Cancel the common factor.

$f (\frac{π}{12}) = 8 - \cos (\frac{2 π}{2 \cdot 2})$

Rewrite the expression.

$f (\frac{π}{12}) = 8 - \cos (\frac{π}{2})$

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

$f (\frac{π}{12}) = 8 - 0$

Multiply $- 1$ by $0$ .

$f (\frac{π}{12}) = 8 + 0$

Add $8$ and $0$ .

$f (\frac{π}{12}) = 8$

The final answer is $8$ .

$8$

Find the point at $x = \frac{π}{4}$ .

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Replace the variable $x$ with $\frac{π}{4}$ in the expression.

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

Simplify the result.

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Simplify each term.

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Combine $3$ and $\frac{π}{4}$ .

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

Combine the numerators over the common denominator.

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

Add $3 π$ and $π$ .

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

Cancel the common factor of $4$ .

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Cancel the common factor.

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

Divide $π$ by $1$ .

$f (\frac{π}{4}) = 8 - \cos (π)$

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.

$f (\frac{π}{4}) = 8 + \cos (0)$

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

$f (\frac{π}{4}) = 8 - (- 1 \cdot 1)$

Multiply $- 1$ by $1$ .

$f (\frac{π}{4}) = 8 + 1$

Multiply $- 1$ by $- 1$ .

$f (\frac{π}{4}) = 8 + 1$

Add $8$ and $1$ .

$f (\frac{π}{4}) = 9$

The final answer is $9$ .

$9$

Find the point at $x = \frac{5 π}{12}$ .

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Replace the variable $x$ with $\frac{5 π}{12}$ in the expression.

$f (\frac{5 π}{12}) = 8 - \cos (3 (\frac{5 π}{12}) + \frac{π}{4})$

Simplify the result.

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Simplify each term.

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Cancel the common factor of $3$ .

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Factor $3$ out of $12$ .

$f (\frac{5 π}{12}) = 8 - \cos (3 (\frac{5 π}{3 (4)}) + \frac{π}{4})$

Cancel the common factor.

$f (\frac{5 π}{12}) = 8 - \cos (3 (\frac{5 π}{3 \cdot 4}) + \frac{π}{4})$

Rewrite the expression.

$f (\frac{5 π}{12}) = 8 - \cos (\frac{5 π}{4} + \frac{π}{4})$

Combine the numerators over the common denominator.

$f (\frac{5 π}{12}) = 8 - \cos (\frac{5 π + π}{4})$

Add $5 π$ and $π$ .

$f (\frac{5 π}{12}) = 8 - \cos (\frac{6 π}{4})$

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

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Factor $2$ out of $6 π$ .

$f (\frac{5 π}{12}) = 8 - \cos (\frac{2 (3 π)}{4})$

Cancel the common factors.

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Factor $2$ out of $4$ .

$f (\frac{5 π}{12}) = 8 - \cos (\frac{2 (3 π)}{2 (2)})$

Cancel the common factor.

$f (\frac{5 π}{12}) = 8 - \cos (\frac{2 (3 π)}{2 \cdot 2})$

Rewrite the expression.

$f (\frac{5 π}{12}) = 8 - \cos (\frac{3 π}{2})$

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

$f (\frac{5 π}{12}) = 8 - \cos (\frac{π}{2})$

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

$f (\frac{5 π}{12}) = 8 - 0$

Multiply $- 1$ by $0$ .

$f (\frac{5 π}{12}) = 8 + 0$

Add $8$ and $0$ .

$f (\frac{5 π}{12}) = 8$

The final answer is $8$ .

$8$

Find the point at $x = \frac{7 π}{12}$ .

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Replace the variable $x$ with $\frac{7 π}{12}$ in the expression.

$f (\frac{7 π}{12}) = 8 - \cos (3 (\frac{7 π}{12}) + \frac{π}{4})$

Simplify the result.

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Simplify each term.

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Cancel the common factor of $3$ .

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Factor $3$ out of $12$ .

$f (\frac{7 π}{12}) = 8 - \cos (3 (\frac{7 π}{3 (4)}) + \frac{π}{4})$

Cancel the common factor.

$f (\frac{7 π}{12}) = 8 - \cos (3 (\frac{7 π}{3 \cdot 4}) + \frac{π}{4})$

Rewrite the expression.

$f (\frac{7 π}{12}) = 8 - \cos (\frac{7 π}{4} + \frac{π}{4})$

Combine the numerators over the common denominator.

$f (\frac{7 π}{12}) = 8 - \cos (\frac{7 π + π}{4})$

Add $7 π$ and $π$ .

$f (\frac{7 π}{12}) = 8 - \cos (\frac{8 π}{4})$

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

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Factor $4$ out of $8 π$ .

$f (\frac{7 π}{12}) = 8 - \cos (\frac{4 (2 π)}{4})$

Cancel the common factors.

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Factor $4$ out of $4$ .

$f (\frac{7 π}{12}) = 8 - \cos (\frac{4 (2 π)}{4 (1)})$

Cancel the common factor.

$f (\frac{7 π}{12}) = 8 - \cos (\frac{4 (2 π)}{4 \cdot 1})$

Rewrite the expression.

$f (\frac{7 π}{12}) = 8 - \cos (\frac{2 π}{1})$

Divide $2 π$ by $1$ .

$f (\frac{7 π}{12}) = 8 - \cos (2 π)$

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

$f (\frac{7 π}{12}) = 8 - \cos (0)$

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

$f (\frac{7 π}{12}) = 8 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (\frac{7 π}{12}) = 8 - 1$

Subtract $1$ from $8$ .

$f (\frac{7 π}{12}) = 7$

The final answer is $7$ .

$7$

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{π}{12} & 7 \\ \frac{π}{12} & 8 \\ \frac{π}{4} & 9 \\ \frac{5 π}{12} & 8 \\ \frac{7 π}{12} & 7 \end{array}$

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

Amplitude: $1$

Period: $\frac{2 π}{3}$

Phase Shift: $- \frac{π}{12}$ ( $\frac{π}{12}$ to the left)

Vertical Shift: $8$

$\begin{array}{c} x & f (x) \\ - \frac{π}{12} & 7 \\ \frac{π}{12} & 8 \\ \frac{π}{4} & 9 \\ \frac{5 π}{12} & 8 \\ \frac{7 π}{12} & 7 \end{array}$

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