Graph y=1+1/2cos(pix)

$y = 1 + \frac{1}{2} \cos (π x)$

Step 1

Rewrite the expression as $\frac{\cos (π x)}{2} + 1$ .

$\frac{\cos (π x)}{2} + 1$

Step 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.

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

$b = π$

$c = 0$

$d = 1$

Step 3

Find the amplitude $| a |$ .

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

Step 4

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

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| π |}$

$π$ is approximately $3.14159265$ which is positive so remove the absolute value

$\frac{2 π}{π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{2 π}{π}$

Divide $2$ by $1$ .

$2$

Find the period of $1$ .

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| π |}$

$π$ is approximately $3.14159265$ which is positive so remove the absolute value

$\frac{2 π}{π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{2 π}{π}$

Divide $2$ by $1$ .

$2$

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

$2$

Step 5

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: $\frac{0}{π}$

Divide $0$ by $π$ .

Phase Shift: $0$

Step 6

List the properties of the trigonometric function.

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

Period: $2$

Phase Shift: None

Vertical Shift: $1$

Step 7

Select a few points to graph.

Find the point at $x = 0$ .

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

$f (0) = 1 + \frac{\cos (π (0))}{2}$

Simplify the result.

Simplify the numerator.

Multiply $π$ by $0$ .

$f (0) = 1 + \frac{\cos (0)}{2}$

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

$f (0) = 1 + \frac{1}{2}$

Simplify the expression.

Write $1$ as a fraction with a common denominator.

$f (0) = \frac{2}{2} + \frac{1}{2}$

Combine the numerators over the common denominator.

$f (0) = \frac{2 + 1}{2}$

Add $2$ and $1$ .

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

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

$\frac{3}{2}$

Find the point at $x = \frac{1}{2}$ .

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

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

Simplify the result.

Simplify each term.

Simplify the numerator.

Combine $π$ and $\frac{1}{2}$ .

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

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

$f (\frac{1}{2}) = 1 + \frac{0}{2}$

Divide $0$ by $2$ .

$f (\frac{1}{2}) = 1 + 0$

Add $1$ and $0$ .

$f (\frac{1}{2}) = 1$

The final answer is $1$ .

$1$

Find the point at $x = 1$ .

Replace the variable $x$ with $1$ in the expression.

$f (1) = 1 + \frac{\cos (π (1))}{2}$

Simplify the result.

Simplify each term.

Simplify the numerator.

Multiply $π$ by $1$ .

$f (1) = 1 + \frac{\cos (π)}{2}$

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.

$f (1) = 1 + \frac{- \cos (0)}{2}$

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

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

Multiply $- 1$ by $1$ .

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

Move the negative in front of the fraction.

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

Simplify the expression.

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Subtract $1$ from $2$ .

$f (1) = \frac{1}{2}$

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

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

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

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

Simplify the result.

Simplify each term.

Simplify the numerator.

Combine $π$ and $\frac{3}{2}$ .

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

Move $3$ to the left of $π$ .

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

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

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

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

$f (\frac{3}{2}) = 1 + \frac{0}{2}$

Divide $0$ by $2$ .

$f (\frac{3}{2}) = 1 + 0$

Add $1$ and $0$ .

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

The final answer is $1$ .

$1$

Find the point at $x = 2$ .

Replace the variable $x$ with $2$ in the expression.

$f (2) = 1 + \frac{\cos (π (2))}{2}$

Simplify the result.

Simplify the numerator.

Move $2$ to the left of $π$ .

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

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

$f (2) = 1 + \frac{\cos (0)}{2}$

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

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

Simplify the expression.

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $2$ and $1$ .

$f (2) = \frac{3}{2}$

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

$\frac{3}{2}$

List the points in a table.

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

Step 8

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

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

Period: $2$

Phase Shift: None

Vertical Shift: $1$

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

Step 9

Graph y=1+1/2cos(pix)

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