Solved on Jan 12, 2024

Graph the cosine function with period $2\pi$ .

STEP 1

Assumptions
1. We are working with the function $y = \cos(2x)$ .
2. The graph will be plotted in a standard Cartesian coordinate system.
3. The domain for $x$ will be chosen to show at least one period of the cosine function.
4. We will identify key points such as maximums, minimums, and intercepts to plot the graph accurately.

STEP 2

Understand the basic shape of the cosine function.
The cosine function has a characteristic wave shape that repeats every $2\pi$ radians. The standard cosine function $y = \cos(x)$ has a maximum value of 1 and a minimum value of -1.

STEP 3

Determine the period of the function $y = \cos(2x)$ .
The period of the standard cosine function is $2\pi$ . For $y = \cos(2x)$ , the period is reduced by a factor of 2. Therefore, the period $P$ is given by:
$P = \frac{2\pi}{2} = \pi$

STEP 4

Identify the amplitude of the function.
The amplitude of $y = \cos(2x)$ is the same as the standard cosine function, which is 1. This means the graph will oscillate between 1 and -1.

STEP 5

Determine the key points for one period of the function.
For the cosine function, key points within one period include:
- The maximum point at the beginning of the period. - The point where the function crosses the x-axis. - The minimum point. - The point where the function crosses the x-axis again. - The maximum point at the end of the period.

STEP 6

Calculate the x-coordinates for the key points.
Since the period is $\pi$ , the key points for $y = \cos(2x)$ within one period are:
- Maximum at $x = 0$ (since $\cos(0) = 1$ ). - Intercept at $x = \frac{\pi}{4}$ (since $\cos(\frac{\pi}{2}) = 0$ ). - Minimum at $x = \frac{\pi}{2}$ (since $\cos(\pi) = -1$ ). - Intercept at $x = \frac{3\pi}{4}$ (since $\cos(\frac{3\pi}{2}) = 0$ ). - Maximum at $x = \pi$ (since $\cos(2\pi) = 1$ ).

STEP 7

Plot the key points on the graph.
- Plot the point (0, 1) for the maximum. - Plot the point ( $\frac{\pi}{4}$ , 0) for the intercept. - Plot the point ( $\frac{\pi}{2}$ , -1) for the minimum. - Plot the point ( $\frac{3\pi}{4}$ , 0) for the intercept. - Plot the point ( $\pi$ , 1) for the maximum.

STEP 8

Draw the curve through the key points.
The graph of $y = \cos(2x)$ should be a smooth wave-like curve that passes through the plotted key points. Ensure that the curve has the shape of a cosine wave, with smooth peaks and troughs.

STEP 9

Extend the graph beyond one period if necessary.
To provide a more complete picture of the function, you can extend the graph by repeating the pattern established in the first period. Remember that the function is periodic with period $\pi$ , so the pattern will repeat every $\pi$ units along the x-axis.
The graph of $y = \cos(2x)$ is now complete. It should show a cosine wave with a period of $\pi$ , oscillating between 1 and -1, with the key points plotted and the curve smoothly connecting them.

Was this helpful?

Graph the cosine function with period $2\pi$ .

STEP 1

STEP 2

Understand the basic shape of the cosine function.
The cosine function has a characteristic wave shape that repeats every $2\pi$ radians. The standard cosine function $y = \cos(x)$ has a maximum value of 1 and a minimum value of -1.

STEP 3

Determine the period of the function $y = \cos(2x)$ .
The period of the standard cosine function is $2\pi$ . For $y = \cos(2x)$ , the period is reduced by a factor of 2. Therefore, the period $P$ is given by:
$P = \frac{2\pi}{2} = \pi$

STEP 4

Identify the amplitude of the function.
The amplitude of $y = \cos(2x)$ is the same as the standard cosine function, which is 1. This means the graph will oscillate between 1 and -1.

STEP 5

STEP 6

STEP 7

Plot the key points on the graph.
- Plot the point (0, 1) for the maximum. - Plot the point ( $\frac{\pi}{4}$ , 0) for the intercept. - Plot the point ( $\frac{\pi}{2}$ , -1) for the minimum. - Plot the point ( $\frac{3\pi}{4}$ , 0) for the intercept. - Plot the point ( $\pi$ , 1) for the maximum.

STEP 8

Draw the curve through the key points.
The graph of $y = \cos(2x)$ should be a smooth wave-like curve that passes through the plotted key points. Ensure that the curve has the shape of a cosine wave, with smooth peaks and troughs.

STEP 9

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Graph the cosine function with period 2π2\pi2π.

STEP 1

STEP 2

Understand the basic shape of the cosine function.The cosine function has a characteristic wave shape that repeats every 2π2\pi2π radians. The standard cosine function y=cos⁡(x)y = \cos(x)y=cos(x) has a maximum value of 1 and a minimum value of -1.

STEP 3

Determine the period of the function y=cos⁡(2x)y = \cos(2x)y=cos(2x).The period of the standard cosine function is 2π2\pi2π. For y=cos⁡(2x)y = \cos(2x)y=cos(2x), the period is reduced by a factor of 2. Therefore, the period PPP is given by:P=2π2=πP = \frac{2\pi}{2} = \piP=22π​=π

STEP 4

Identify the amplitude of the function.The amplitude of y=cos⁡(2x)y = \cos(2x)y=cos(2x) is the same as the standard cosine function, which is 1. This means the graph will oscillate between 1 and -1.

STEP 5

STEP 6

STEP 7

STEP 8

Draw the curve through the key points.The graph of y=cos⁡(2x)y = \cos(2x)y=cos(2x) should be a smooth wave-like curve that passes through the plotted key points. Ensure that the curve has the shape of a cosine wave, with smooth peaks and troughs.

STEP 9

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Graph the cosine function with period $2\pi$ .

Understand the basic shape of the cosine function.
The cosine function has a characteristic wave shape that repeats every $2\pi$ radians. The standard cosine function $y = \cos(x)$ has a maximum value of 1 and a minimum value of -1.

Determine the period of the function $y = \cos(2x)$ .
The period of the standard cosine function is $2\pi$ . For $y = \cos(2x)$ , the period is reduced by a factor of 2. Therefore, the period $P$ is given by:
$P = \frac{2\pi}{2} = \pi$

Identify the amplitude of the function.
The amplitude of $y = \cos(2x)$ is the same as the standard cosine function, which is 1. This means the graph will oscillate between 1 and -1.

Draw the curve through the key points.
The graph of $y = \cos(2x)$ should be a smooth wave-like curve that passes through the plotted key points. Ensure that the curve has the shape of a cosine wave, with smooth peaks and troughs.