Set your graph options
The default settings draw y = cos(2x).
Example data table
For y = cos(2x), these familiar radian inputs show one full cycle.
| x | 2x | cos(2x) | Graph observation |
|---|---|---|---|
| 0 | 0 | 1 | Starts at a maximum. |
| π/4 | π/2 | 0 | Crosses the midline. |
| π/2 | π | -1 | Reaches the minimum. |
| 3π/4 | 3π/2 | 0 | Crosses upward again. |
| π | 2π | 1 | Completes one cycle. |
Formula used
y = A cos(Bx + C) + D
For the standard graph, A = 1, B = 2, C = 0, and D = 0. Therefore, y = cos(2x). The amplitude is |A|. The period in radians is 2π / |B|. For cos(2x), the period equals π.
The value C changes phase within the angle. The value D shifts the midline. A negative amplitude reflects the curve across its midline.
How to use this calculator
- Keep the default values to graph the standard cos(2x) function.
- Adjust amplitude, frequency, phase, or vertical shift when required.
- Set a useful x-range and a positive step size.
- Select radians or degrees before calculating the curve.
- Press Calculate and graph to view the result above.
- Download the listed values as CSV or save a PDF report.
Understand the cos 2x graph
A faster cosine cycle
The graph y = cos(2x) is a cosine curve. Its coefficient changes the horizontal rate. The number 2 doubles the angle before cosine evaluates it. This makes the curve complete cycles faster. A usual cosine graph has period 2π. The cos(2x) graph has period π. It fits two complete cycles within 0 to 2π.
Key points make graphing easier
Start at x = 0. The value is cos(0), so y equals 1. At x = π/4, the doubled angle becomes π/2. The graph crosses its midline there. At x = π/2, the doubled angle is π. The value reaches -1. These points reveal the full wave pattern. Repeat the pattern every π units.
Read transformations clearly
The broader form is y = A cos(Bx + C) + D. Amplitude A changes vertical height. A value of 3 creates peaks three units away from the midline. Multiplier B controls the period. Larger absolute B values shorten the cycle. Phase value C moves the wave horizontally. Shift D moves the entire graph vertically. The calculator displays these effects immediately.
Choose a useful graph range
Radians are common in algebra and calculus courses. A range from -2π to 2π shows four cycles for cos(2x). Use a small step size for a smooth plot. A larger step creates fewer table rows. Degree mode can help with school exercises. In degrees, cos(2x) repeats every 180 degrees.
Check your output
Review the maximum, minimum, midline, and period after calculation. The basic function has maximum 1 and minimum -1. Its midline is y = 0. Exporting a CSV helps with spreadsheets. The PDF option creates a compact report for study notes. Compare the optional reference line when testing transformed functions. Accurate inputs produce reliable graph interpretations.
Practice with settings before relying on memory. Begin with the function, then change one parameter at a time. Increase amplitude, alter the frequency, or move the midline. Notice which feature changes after each calculation. Write down the period and intercepts for every trial. This comparison improves pattern recognition. It also prepares you for equations written in unfamiliar forms. The graph and table work together. One offers a visual model. The other provides numerical evidence. Use both when checking homework, preparing lessons, or reviewing trigonometric transformations carefully.
Frequently asked questions
What does cos(2x) mean?
It means cosine is evaluated after x is multiplied by two. The doubled angle compresses the usual cosine curve horizontally.
What is the period of cos(2x)?
Its period is π radians, or 180 degrees. The multiplier 2 halves the usual cosine period of 2π.
What is the amplitude of cos(2x)?
The amplitude is 1. The values remain between -1 and 1 because no outside multiplier changes the height.
Where does cos(2x) reach a maximum?
It reaches 1 when 2x equals 2πk, where k is any integer. Equivalently, x equals πk.
Where does cos(2x) reach a minimum?
It reaches -1 when 2x equals π plus 2πk. This gives x = π/2 + πk.
Where are the zeros of cos(2x)?
The zeros occur when 2x equals π/2 plus πk. Therefore, x = π/4 + πk/2.
Can I graph in degrees?
Yes. Choose Degrees before calculation. The calculator converts degree inputs internally, then returns labels in degrees.
Why is a smaller step size useful?
A smaller step adds more computed points. More points make the plotted curve smoother and improve detail near important locations.
How does a negative amplitude change the graph?
A negative amplitude reflects the cosine curve across its midline. The period stays unchanged unless the frequency multiplier also changes.
What does the vertical shift do?
The vertical shift changes the midline. For example, adding 3 moves every cosine value upward by three units.
Can I export the calculated values?
Yes. After calculation, use Download CSV for data analysis or Download PDF for a printable graph report.