Graph y=cos(x+45 degrees )

$y = \cos (x + 45 °)$

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.

$a = 1$

$b = 1$

$c = - 45 °$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

Find the period of $\cos (x + 45 °)$ .

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 $1$ .

$2 π$

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: $\frac{- 45 °}{1}$

Divide $- 45 °$ by $1$ .

Phase Shift: $- 45$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $2 π$

Phase Shift: $- 45$ ( $45$ to the left)

Vertical Shift: None

Step 6

Select a few points to graph.

Find the point at $x = - 45$ .

Replace the variable $x$ with $- 45$ in the expression.

$f (- 45) = \cos ((- 45) + 45 °)$

Simplify the result.

Add $- 45$ and $45 °$ .

$f (- 45) = \cos (0)$

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

$f (- 45) = 1$

The final answer is $1$ .

$1$

Find the point at $x = \frac{π}{2} - 45$ .

Replace the variable $x$ with $\frac{π}{2} - 45$ in the expression.

$f (\frac{π}{2} - 45) = \cos ((\frac{π}{2} - 45) + 45 °)$

Simplify the result.

Add $- 45$ and $45 °$ .

$f (\frac{π}{2} - 45) = \cos (\frac{π}{2} + 0)$

Add $\frac{π}{2}$ and $0$ .

$f (\frac{π}{2} - 45) = \cos (\frac{π}{2})$

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

$f (\frac{π}{2} - 45) = 0$

The final answer is $0$ .

$0$

Find the point at $x = π - 45$ .

Replace the variable $x$ with $π - 45$ in the expression.

$f (π - 45) = \cos ((π - 45) + 45 °)$

Simplify the result.

Add $- 45$ and $45 °$ .

$f (π - 45) = \cos (π + 0)$

Add $π$ and $0$ .

$f (π - 45) = \cos (π)$

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.

$f (π - 45) = - \cos (0)$

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

$f (π - 45) = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (π - 45) = - 1$

The final answer is $- 1$ .

$- 1$

Find the point at $x = \frac{3 π}{2} - 45$ .

Replace the variable $x$ with $\frac{3 π}{2} - 45$ in the expression.

$f (\frac{3 π}{2} - 45) = \cos ((\frac{3 π}{2} - 45) + 45 °)$

Simplify the result.

Add $- 45$ and $45 °$ .

$f (\frac{3 π}{2} - 45) = \cos (\frac{3 π}{2} + 0)$

Add $\frac{3 π}{2}$ and $0$ .

$f (\frac{3 π}{2} - 45) = \cos (\frac{3 π}{2})$

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

$f (\frac{3 π}{2} - 45) = \cos (\frac{π}{2})$

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

$f (\frac{3 π}{2} - 45) = 0$

The final answer is $0$ .

$0$

Find the point at $x = 2 π - 45$ .

Replace the variable $x$ with $2 π - 45$ in the expression.

$f (2 π - 45) = \cos ((2 π - 45) + 45 °)$

Simplify the result.

Add $- 45$ and $45 °$ .

$f (2 π - 45) = \cos (2 π + 0)$

Add $2 π$ and $0$ .

$f (2 π - 45) = \cos (2 π)$

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

$f (2 π - 45) = \cos (0)$

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

$f (2 π - 45) = 1$

The final answer is $1$ .

$1$

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 45 & 1 \\ \frac{π}{2} - 45 & 0 \\ π - 45 & - 1 \\ \frac{3 π}{2} - 45 & 0 \\ 2 π - 45 & 1 \end{array}$

Step 7

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

Amplitude: $1$

Period: $2 π$

Phase Shift: $- 45$ ( $45$ to the left)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - 45 & 1 \\ \frac{π}{2} - 45 & 0 \\ π - 45 & - 1 \\ \frac{3 π}{2} - 45 & 0 \\ 2 π - 45 & 1 \end{array}$

Step 8

Graph y=cos(x+45 degrees )

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