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

$y = \frac{1}{2} \cos (x + \frac{π}{2}) + 2$

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

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

$b = 1$

$c = - \frac{π}{2}$

$d = 2$

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{2}$

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

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

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 period of $2$ .

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 $1$ .

$2 π$

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

$2 π$

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: $\frac{- \frac{π}{2}}{1}$

Divide $- \frac{π}{2}$ by $1$ .

Phase Shift: $- \frac{π}{2}$

List the properties of the trigonometric function.

Amplitude: $\frac{1}{2}$

Period: $2 π$

Phase Shift: $- \frac{π}{2}$ ( $\frac{π}{2}$ to the left)

Vertical Shift: $2$

Select a few points to graph.

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

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

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

Simplify the result.

Add $- \frac{π}{2}$ and $\frac{π}{2}$ .

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

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

$f (- \frac{π}{2}) = \frac{1}{2} + 2$

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $2$ .

$f (- \frac{π}{2}) = \frac{1 + 4}{2}$

Add $1$ and $4$ .

$f (- \frac{π}{2}) = \frac{5}{2}$

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

$\frac{5}{2}$

Find the point at $x = 0$ .

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

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

Simplify the result.

Add $0$ and $\frac{π}{2}$ .

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

Simplify each term.

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

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

Divide $0$ by $2$ .

$f (0) = 0 + 2$

Add $0$ and $2$ .

$f (0) = 2$

The final answer is $2$ .

$2$

Find the point at $x = \frac{π}{2}$ .

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

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

Simplify the result.

Simplify by adding terms.

Combine the numerators over the common denominator.

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

Add $π$ and $π$ .

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

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $π$ by $1$ .

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

Simplify the numerator.

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.

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

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

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

Multiply $- 1$ by $1$ .

$f (\frac{π}{2}) = \frac{- 1}{2} + 2$

Move the negative in front of the fraction.

$f (\frac{π}{2}) = - \frac{1}{2} + 2$

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $2$ .

$f (\frac{π}{2}) = \frac{- 1 + 4}{2}$

Add $- 1$ and $4$ .

$f (\frac{π}{2}) = \frac{3}{2}$

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

$\frac{3}{2}$

Find the point at $x = π$ .

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

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

Simplify the result.

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

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

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Add $π \cdot 2$ and $π$ .

Reorder $π$ and $2$ .

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

Add $2 \cdot π$ and $π$ .

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

Simplify each term.

Simplify the numerator.

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

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

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

$f (π) = \frac{0}{2} + 2$

Divide $0$ by $2$ .

$f (π) = 0 + 2$

Add $0$ and $2$ .

$f (π) = 2$

The final answer is $2$ .

$2$

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

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

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

Simplify the result.

Simplify by adding terms.

Combine the numerators over the common denominator.

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

Add $3 π$ and $π$ .

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

Simplify each term.

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

Factor $2$ out of $4 π$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2 π$ by $1$ .

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

Simplify the numerator.

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

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

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

$f (\frac{3 π}{2}) = \frac{1}{2} + 2$

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $2$ .

$f (\frac{3 π}{2}) = \frac{1 + 4}{2}$

Add $1$ and $4$ .

$f (\frac{3 π}{2}) = \frac{5}{2}$

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

$\frac{5}{2}$

List the points in a table.

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

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

Amplitude: $\frac{1}{2}$

Period: $2 π$

Phase Shift: $- \frac{π}{2}$ ( $\frac{π}{2}$ to the left)

Vertical Shift: $2$

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

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

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