Graph y=2cos(pi(x+1/4))

$y = 2 \cos (π (x + \frac{1}{4}))$

Step 1

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

$a = 2$

$b = π$

$c = - \frac{π}{4}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $2$

Step 3

Find the period of $2 \cos (π x + \frac{π}{4})$ .

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 $1$ .

$2$

Step 4

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: $\frac{- \frac{π}{4}}{π}$

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $π$ .

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

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

Factor $π$ out of $- π$ .

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

Cancel the common factor.

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

Rewrite the expression.

Phase Shift: $\frac{- 1}{4}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{1}{4}$

Step 5

List the properties of the trigonometric function.

Amplitude: $2$

Period: $2$

Phase Shift: $- \frac{1}{4}$ ( $\frac{1}{4}$ to the left)

Vertical Shift: None

Step 6

Select a few points to graph.

Find the point at $x = - \frac{1}{4}$ .

Replace the variable $x$ with $- \frac{1}{4}$ in the expression.

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

Simplify the result.

Combine $π$ and $\frac{1}{4}$ .

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

Combine the numerators over the common denominator.

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

Add $- π$ and $π$ .

$f (- \frac{1}{4}) = 2 \cos (\frac{0}{4})$

Divide $0$ by $4$ .

$f (- \frac{1}{4}) = 2 \cos (0)$

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

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

Multiply $2$ by $1$ .

$f (- \frac{1}{4}) = 2$

The final answer is $2$ .

$2$

Find the point at $x = \frac{1}{4}$ .

Replace the variable $x$ with $\frac{1}{4}$ in the expression.

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

Simplify the result.

Combine $π$ and $\frac{1}{4}$ .

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

Combine the numerators over the common denominator.

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

Add $π$ and $π$ .

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

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

Factor $2$ out of $2 π$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

$f (\frac{1}{4}) = 2 \cos (\frac{2 π}{2 \cdot 2})$

Cancel the common factor.

$f (\frac{1}{4}) = 2 \cos (\frac{2 π}{2 \cdot 2})$

Rewrite the expression.

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

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

$f (\frac{1}{4}) = 2 \cdot 0$

Multiply $2$ by $0$ .

$f (\frac{1}{4}) = 0$

The final answer is $0$ .

$0$

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

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

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

Simplify the result.

Simplify each term.

Combine $π$ and $\frac{3}{4}$ .

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

Move $3$ to the left of $π$ .

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

Simplify terms.

Combine the numerators over the common denominator.

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

Add $3 π$ and $π$ .

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $π$ by $1$ .

$f (\frac{3}{4}) = 2 \cos (π)$

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.

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

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

$f (\frac{3}{4}) = 2 (- 1 \cdot 1)$

Multiply $2 (- 1 \cdot 1)$ .

Multiply $- 1$ by $1$ .

$f (\frac{3}{4}) = 2 \cdot - 1$

Multiply $2$ by $- 1$ .

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

The final answer is $- 2$ .

$- 2$

Find the point at $x = \frac{5}{4}$ .

Replace the variable $x$ with $\frac{5}{4}$ in the expression.

$f (\frac{5}{4}) = 2 \cos (π (\frac{5}{4}) + \frac{π}{4})$

Simplify the result.

Simplify each term.

Combine $π$ and $\frac{5}{4}$ .

$f (\frac{5}{4}) = 2 \cos (\frac{π \cdot 5}{4} + \frac{π}{4})$

Move $5$ to the left of $π$ .

$f (\frac{5}{4}) = 2 \cos (\frac{5 π}{4} + \frac{π}{4})$

Simplify terms.

Combine the numerators over the common denominator.

$f (\frac{5}{4}) = 2 \cos (\frac{5 π + π}{4})$

Add $5 π$ and $π$ .

$f (\frac{5}{4}) = 2 \cos (\frac{6 π}{4})$

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

Factor $2$ out of $6 π$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

$f (\frac{5}{4}) = 2 \cos (\frac{2 (3 π)}{2 \cdot 2})$

Rewrite the expression.

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

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

$f (\frac{5}{4}) = 2 \cos (\frac{π}{2})$

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

$f (\frac{5}{4}) = 2 \cdot 0$

Multiply $2$ by $0$ .

$f (\frac{5}{4}) = 0$

The final answer is $0$ .

$0$

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

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

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

Simplify the result.

Simplify each term.

Combine $π$ and $\frac{7}{4}$ .

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

Move $7$ to the left of $π$ .

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

Simplify terms.

Combine the numerators over the common denominator.

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

Add $7 π$ and $π$ .

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

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

Factor $4$ out of $8 π$ .

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

Cancel the common factors.

Factor $4$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2 π$ by $1$ .

$f (\frac{7}{4}) = 2 \cos (2 π)$

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

$f (\frac{7}{4}) = 2 \cos (0)$

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

$f (\frac{7}{4}) = 2 \cdot 1$

Multiply $2$ by $1$ .

$f (\frac{7}{4}) = 2$

The final answer is $2$ .

$2$

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{1}{4} & 2 \\ \frac{1}{4} & 0 \\ \frac{3}{4} & - 2 \\ \frac{5}{4} & 0 \\ \frac{7}{4} & 2 \end{array}$

Step 7

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

Amplitude: $2$

Period: $2$

Phase Shift: $- \frac{1}{4}$ ( $\frac{1}{4}$ to the left)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - \frac{1}{4} & 2 \\ \frac{1}{4} & 0 \\ \frac{3}{4} & - 2 \\ \frac{5}{4} & 0 \\ \frac{7}{4} & 2 \end{array}$

Step 8

Graph y=2cos(pi(x+1/4))

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