Graph f(x)=4-2cos(4x+pi)

$f (x) = 4 - 2 \cos (4 x + π)$

Rewrite the expression as $- 2 \cos (4 x + π) + 4$ .

$- 2 \cos (4 x + π) + 4$

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

$a = - 2$

$b = 4$

$c = - π$

$d = 4$

Find the amplitude $| a |$ .

Amplitude: $2$

Find the period of $- 2 \cos (4 x + π)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

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

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Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Cancel the common factors.

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Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$\frac{π}{2}$

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: $\frac{- π}{4}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{4}$

Find the vertical shift $d$ .

Vertical Shift: $4$

List the properties of the trigonometric function.

Amplitude: $2$

Period: $\frac{π}{2}$

Phase Shift: $- \frac{π}{4}$ ( $\frac{π}{4}$ to the left)

Vertical Shift: $4$

Select a few points to graph.

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Find the point at $x = - \frac{π}{4}$ .

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

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

Simplify the result.

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Simplify each term.

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Cancel the common factor of $4$ .

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Move the leading negative in $- \frac{π}{4}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Add $- π$ and $π$ .

$f (- \frac{π}{4}) = 4 - 2 \cos (0)$

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

$f (- \frac{π}{4}) = 4 - 2 \cdot 1$

Multiply $- 2$ by $1$ .

$f (- \frac{π}{4}) = 4 - 2$

Subtract $2$ from $4$ .

$f (- \frac{π}{4}) = 2$

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.

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

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 $4$ .

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

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

Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Move $2$ to the left of $π$ .

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

Add $- π$ and $2 π$ .

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

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

$f (- \frac{π}{8}) = 4 - 2 \cdot 0$

Multiply $- 2$ by $0$ .

$f (- \frac{π}{8}) = 4 + 0$

Add $4$ and $0$ .

$f (- \frac{π}{8}) = 4$

The final answer is $4$ .

$4$

Find the point at $x = 0$ .

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

$f (0) = 4 - 2 \cos (4 (0) + π)$

Simplify the result.

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Simplify each term.

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Multiply $4$ by $0$ .

$f (0) = 4 - 2 \cos (0 + π)$

Add $0$ and $π$ .

$f (0) = 4 - 2 \cos (π)$

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.

$f (0) = 4 - 2 (- \cos (0))$

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

$f (0) = 4 - 2 (- 1 \cdot 1)$

Multiply $- 1$ by $1$ .

$f (0) = 4 - 2 \cdot - 1$

Multiply $- 2$ by $- 1$ .

$f (0) = 4 + 2$

Add $4$ and $2$ .

$f (0) = 6$

The final answer is $6$ .

$6$

Find the point at $x = \frac{π}{8}$ .

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

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

Simplify the result.

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Simplify each term.

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Cancel the common factor of $4$ .

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Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Move $2$ to the left of $π$ .

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

Add $π$ and $2 π$ .

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

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

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

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

$f (\frac{π}{8}) = 4 - 2 \cdot 0$

Multiply $- 2$ by $0$ .

$f (\frac{π}{8}) = 4 + 0$

Add $4$ and $0$ .

$f (\frac{π}{8}) = 4$

The final answer is $4$ .

$4$

Find the point at $x = \frac{π}{4}$ .

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

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

Simplify the result.

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Simplify each term.

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Cancel the common factor of $4$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Add $π$ and $π$ .

$f (\frac{π}{4}) = 4 - 2 \cos (2 π)$

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

$f (\frac{π}{4}) = 4 - 2 \cos (0)$

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

$f (\frac{π}{4}) = 4 - 2 \cdot 1$

Multiply $- 2$ by $1$ .

$f (\frac{π}{4}) = 4 - 2$

Subtract $2$ from $4$ .

$f (\frac{π}{4}) = 2$

The final answer is $2$ .

$2$

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{π}{4} & 2 \\ - \frac{π}{8} & 4 \\ 0 & 6 \\ \frac{π}{8} & 4 \\ \frac{π}{4} & 2 \end{array}$

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

Amplitude: $2$

Period: $\frac{π}{2}$

Phase Shift: $- \frac{π}{4}$ ( $\frac{π}{4}$ to the left)

Vertical Shift: $4$

$\begin{array}{c} x & f (x) \\ - \frac{π}{4} & 2 \\ - \frac{π}{8} & 4 \\ 0 & 6 \\ \frac{π}{8} & 4 \\ \frac{π}{4} & 2 \end{array}$

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