Graph y=-2cos(x/2-(3pi)/8)+2

$y = - 2 \cos (\frac{x}{2} - \frac{3 π}{8}) + 2$

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

$a = - 2$

$b = \frac{1}{2}$

$c = \frac{3 π}{8}$

$d = 2$

Find the amplitude $| a |$ .

Amplitude: $2$

Find the period using the formula $\frac{2 π}{| b |}$ .

Find the period of $- 2 \cos (\frac{x}{2} - \frac{3 π}{8})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Multiply $2$ by $2$ .

$4 π$

Find the period of $2$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Multiply $2$ by $2$ .

$4 π$

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$4 π$

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

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: $\frac{\frac{3 π}{8}}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{3 π}{8} \cdot 2$

Cancel the common factor of $2$ .

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

Phase Shift: $\frac{3 π}{4}$

List the properties of the trigonometric function.

Amplitude: $2$

Period: $4 π$

Phase Shift: $\frac{3 π}{4}$ ( $\frac{3 π}{4}$ to the right)

Vertical Shift: $2$

Select a few points to graph.

Find the point at $x = \frac{3 π}{4}$ .

Replace the variable $x$ with $\frac{3 π}{4}$ in the expression.

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

Simplify the result.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Multiply $4$ by $2$ .

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

Combine the numerators over the common denominator.

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

Subtract $3 π$ from $3 π$ .

$f (\frac{3 π}{4}) = - 2 \cos (\frac{0}{8}) + 2$

Divide $0$ by $8$ .

$f (\frac{3 π}{4}) = - 2 \cos (0) + 2$

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

$f (\frac{3 π}{4}) = - 2 \cdot 1 + 2$

Multiply $- 2$ by $1$ .

$f (\frac{3 π}{4}) = - 2 + 2$

Add $- 2$ and $2$ .

$f (\frac{3 π}{4}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{7 π}{4}$ .

Replace the variable $x$ with $\frac{7 π}{4}$ in the expression.

$f (\frac{7 π}{4}) = - 2 \cos (\frac{\frac{7 π}{4}}{2} - \frac{3 π}{8}) + 2$

Simplify the result.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7 π}{4}) = - 2 \cos (\frac{7 π}{4} \cdot \frac{1}{2} - \frac{3 π}{8}) + 2$

Multiply $\frac{7 π}{4} \cdot \frac{1}{2}$ .

Multiply $\frac{7 π}{4}$ by $\frac{1}{2}$ .

$f (\frac{7 π}{4}) = - 2 \cos (\frac{7 π}{4 \cdot 2} - \frac{3 π}{8}) + 2$

Multiply $4$ by $2$ .

$f (\frac{7 π}{4}) = - 2 \cos (\frac{7 π}{8} - \frac{3 π}{8}) + 2$

Combine the numerators over the common denominator.

$f (\frac{7 π}{4}) = - 2 \cos (\frac{7 π - 3 π}{8}) + 2$

Subtract $3 π$ from $7 π$ .

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

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

Factor $4$ out of $4 π$ .

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

Cancel the common factors.

Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

$f (\frac{7 π}{4}) = - 2 \cdot 0 + 2$

Multiply $- 2$ by $0$ .

$f (\frac{7 π}{4}) = 0 + 2$

Add $0$ and $2$ .

$f (\frac{7 π}{4}) = 2$

The final answer is $2$ .

$2$

Find the point at $x = \frac{11 π}{4}$ .

Replace the variable $x$ with $\frac{11 π}{4}$ in the expression.

$f (\frac{11 π}{4}) = - 2 \cos (\frac{\frac{11 π}{4}}{2} - \frac{3 π}{8}) + 2$

Simplify the result.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{11 π}{4}) = - 2 \cos (\frac{11 π}{4} \cdot \frac{1}{2} - \frac{3 π}{8}) + 2$

Multiply $\frac{11 π}{4} \cdot \frac{1}{2}$ .

Multiply $\frac{11 π}{4}$ by $\frac{1}{2}$ .

$f (\frac{11 π}{4}) = - 2 \cos (\frac{11 π}{4 \cdot 2} - \frac{3 π}{8}) + 2$

Multiply $4$ by $2$ .

$f (\frac{11 π}{4}) = - 2 \cos (\frac{11 π}{8} - \frac{3 π}{8}) + 2$

Combine the numerators over the common denominator.

$f (\frac{11 π}{4}) = - 2 \cos (\frac{11 π - 3 π}{8}) + 2$

Subtract $3 π$ from $11 π$ .

$f (\frac{11 π}{4}) = - 2 \cos (\frac{8 π}{8}) + 2$

Cancel the common factor of $8$ .

Cancel the common factor.

$f (\frac{11 π}{4}) = - 2 \cos (\frac{8 π}{8}) + 2$

Divide $π$ by $1$ .

$f (\frac{11 π}{4}) = - 2 \cos (π) + 2$

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.

$f (\frac{11 π}{4}) = - 2 (- \cos (0)) + 2$

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

$f (\frac{11 π}{4}) = - 2 (- 1 \cdot 1) + 2$

Multiply $- 1$ by $1$ .

$f (\frac{11 π}{4}) = - 2 \cdot - 1 + 2$

Multiply $- 2$ by $- 1$ .

$f (\frac{11 π}{4}) = 2 + 2$

Add $2$ and $2$ .

$f (\frac{11 π}{4}) = 4$

The final answer is $4$ .

$4$

Find the point at $x = \frac{15 π}{4}$ .

Replace the variable $x$ with $\frac{15 π}{4}$ in the expression.

$f (\frac{15 π}{4}) = - 2 \cos (\frac{\frac{15 π}{4}}{2} - \frac{3 π}{8}) + 2$

Simplify the result.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{15 π}{4}) = - 2 \cos (\frac{15 π}{4} \cdot \frac{1}{2} - \frac{3 π}{8}) + 2$

Multiply $\frac{15 π}{4} \cdot \frac{1}{2}$ .

Multiply $\frac{15 π}{4}$ by $\frac{1}{2}$ .

$f (\frac{15 π}{4}) = - 2 \cos (\frac{15 π}{4 \cdot 2} - \frac{3 π}{8}) + 2$

Multiply $4$ by $2$ .

$f (\frac{15 π}{4}) = - 2 \cos (\frac{15 π}{8} - \frac{3 π}{8}) + 2$

Combine the numerators over the common denominator.

$f (\frac{15 π}{4}) = - 2 \cos (\frac{15 π - 3 π}{8}) + 2$

Subtract $3 π$ from $15 π$ .

$f (\frac{15 π}{4}) = - 2 \cos (\frac{12 π}{8}) + 2$

Cancel the common factor of $12$ and $8$ .

Factor $4$ out of $12 π$ .

$f (\frac{15 π}{4}) = - 2 \cos (\frac{4 (3 π)}{8}) + 2$

Cancel the common factors.

Factor $4$ out of $8$ .

$f (\frac{15 π}{4}) = - 2 \cos (\frac{4 (3 π)}{4 (2)}) + 2$

Cancel the common factor.

$f (\frac{15 π}{4}) = - 2 \cos (\frac{4 (3 π)}{4 \cdot 2}) + 2$

Rewrite the expression.

$f (\frac{15 π}{4}) = - 2 \cos (\frac{3 π}{2}) + 2$

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

$f (\frac{15 π}{4}) = - 2 \cos (\frac{π}{2}) + 2$

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

$f (\frac{15 π}{4}) = - 2 \cdot 0 + 2$

Multiply $- 2$ by $0$ .

$f (\frac{15 π}{4}) = 0 + 2$

Add $0$ and $2$ .

$f (\frac{15 π}{4}) = 2$

The final answer is $2$ .

$2$

Find the point at $x = \frac{19 π}{4}$ .

Replace the variable $x$ with $\frac{19 π}{4}$ in the expression.

$f (\frac{19 π}{4}) = - 2 \cos (\frac{\frac{19 π}{4}}{2} - \frac{3 π}{8}) + 2$

Simplify the result.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{19 π}{4}) = - 2 \cos (\frac{19 π}{4} \cdot \frac{1}{2} - \frac{3 π}{8}) + 2$

Multiply $\frac{19 π}{4} \cdot \frac{1}{2}$ .

Multiply $\frac{19 π}{4}$ by $\frac{1}{2}$ .

$f (\frac{19 π}{4}) = - 2 \cos (\frac{19 π}{4 \cdot 2} - \frac{3 π}{8}) + 2$

Multiply $4$ by $2$ .

$f (\frac{19 π}{4}) = - 2 \cos (\frac{19 π}{8} - \frac{3 π}{8}) + 2$

Combine the numerators over the common denominator.

$f (\frac{19 π}{4}) = - 2 \cos (\frac{19 π - 3 π}{8}) + 2$

Subtract $3 π$ from $19 π$ .

$f (\frac{19 π}{4}) = - 2 \cos (\frac{16 π}{8}) + 2$

Cancel the common factor of $16$ and $8$ .

Factor $8$ out of $16 π$ .

$f (\frac{19 π}{4}) = - 2 \cos (\frac{8 (2 π)}{8}) + 2$

Cancel the common factors.

Factor $8$ out of $8$ .

$f (\frac{19 π}{4}) = - 2 \cos (\frac{8 (2 π)}{8 (1)}) + 2$

Cancel the common factor.

$f (\frac{19 π}{4}) = - 2 \cos (\frac{8 (2 π)}{8 \cdot 1}) + 2$

Rewrite the expression.

$f (\frac{19 π}{4}) = - 2 \cos (\frac{2 π}{1}) + 2$

Divide $2 π$ by $1$ .

$f (\frac{19 π}{4}) = - 2 \cos (2 π) + 2$

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

$f (\frac{19 π}{4}) = - 2 \cos (0) + 2$

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

$f (\frac{19 π}{4}) = - 2 \cdot 1 + 2$

Multiply $- 2$ by $1$ .

$f (\frac{19 π}{4}) = - 2 + 2$

Add $- 2$ and $2$ .

$f (\frac{19 π}{4}) = 0$

The final answer is $0$ .

$0$

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{3 π}{4} & 0 \\ \frac{7 π}{4} & 2 \\ \frac{11 π}{4} & 4 \\ \frac{15 π}{4} & 2 \\ \frac{19 π}{4} & 0 \end{array}$

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

Amplitude: $2$

Period: $4 π$

Phase Shift: $\frac{3 π}{4}$ ( $\frac{3 π}{4}$ to the right)

Vertical Shift: $2$

$\begin{array}{c} x & f (x) \\ \frac{3 π}{4} & 0 \\ \frac{7 π}{4} & 2 \\ \frac{11 π}{4} & 4 \\ \frac{15 π}{4} & 2 \\ \frac{19 π}{4} & 0 \end{array}$

Graph y=-2cos(x/2-(3pi)/8)+2

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