Formula Used
For a trigonometric function y = A sin(Bx + C) + D or y = A cos(Bx + C) + D, the amplitude is:
Amplitude = |A|
The range is D - |A| ≤ y ≤ D + |A|. The midline is y = D. The period is 2π / |B| in radians or 360 / |B| in degrees. The phase shift is -C / B.
When maximum and minimum values are known, use:
Amplitude = (maximum - minimum) / 2
Midline = (maximum + minimum) / 2
How to Use This Calculator
- Select the calculation method that matches your problem.
- For a sine or cosine equation, enter A, B, C, and D.
- For extrema, enter the highest and lowest function values.
- For observed values, paste the data series into the text box.
- Choose decimal places and graph limits.
- Press the calculate button to view results above the form.
- Use the CSV or PDF button to save the result.
Example Data Table
| Example |
Input Type |
Values |
Amplitude |
Midline |
| Trig model |
y = -4 sin(2x + 1) + 3 |
A = -4, B = 2, C = 1, D = 3 |
4 |
y = 3 |
| Known extrema |
Maximum and minimum |
Max = 12, Min = -2 |
7 |
y = 5 |
| Observed values |
Data set |
3, 8, 10, 4, -2 |
6 |
y = 4 |
Understanding Function Amplitude
What Amplitude Means
Amplitude measures the vertical size of an oscillating function. It tells how far the graph rises above its midline. It also tells how far the graph falls below that same center line. For sine and cosine curves, amplitude is always a nonnegative value. A negative leading coefficient reflects the graph. It does not make amplitude negative.
Why It Matters
Amplitude is useful in algebra, trigonometry, physics, signal study, and wave motion. It describes height, strength, or variation. A sound wave with larger amplitude is usually louder. A tide model with larger amplitude shows a bigger change between high and low water. A temperature cycle with larger amplitude shows stronger daily movement.
Reading It From an Equation
In the function y = A sin(Bx + C) + D, the amplitude is the absolute value of A. The value B changes the period. The value C shifts the graph left or right. The value D moves the midline up or down. This calculator separates those parts, so the result is easier to understand.
Using Maximum and Minimum Values
Sometimes you do not have the full equation. You may only know the highest and lowest outputs. In that case, subtract the minimum from the maximum. Then divide by two. The same idea works for observed data. It gives an estimated amplitude based on the visible spread of the values.
Common Mistakes
Many learners confuse amplitude with period, range, or vertical shift. These are related, but they are not the same. Amplitude is only the distance from the midline to a peak. The range uses both the midline and amplitude. The period measures one full horizontal cycle. The phase shift moves the curve sideways.
Graph and Export Options
The chart helps you see the amplitude visually. The top and bottom of the wave show the range. The center shows the midline. Export options help save classroom work, reports, or study notes. Use more graph points for smoother curves. Use fewer decimal places when you need simple answers.
Best Practice
Check units before comparing answers. Radians and degrees change period values, but they do not change amplitude result.
Frequently Asked Questions
1. What is the amplitude of a function?
Amplitude is the distance from the midline to the maximum or minimum of an oscillating function. For sine and cosine functions, it equals the absolute value of the leading coefficient A.
2. Can amplitude be negative?
No. Amplitude is a distance, so it is never negative. If A is negative, the graph is reflected across the midline, but the amplitude is still |A|.
3. How do I find amplitude from maximum and minimum?
Subtract the minimum value from the maximum value. Then divide the result by two. The formula is amplitude = (maximum - minimum) / 2.
4. Does vertical shift affect amplitude?
No. Vertical shift moves the whole graph up or down. It changes the midline and range values, but it does not change the amplitude.
5. Does period affect amplitude?
No. Period controls horizontal cycle length. Amplitude controls vertical height from the midline. They describe different parts of the graph.
6. Can this calculator use data values?
Yes. Enter observed values in the data box. The calculator finds the highest and lowest values, then estimates amplitude from half of their difference.
7. What is the midline?
The midline is the horizontal center of the oscillation. For y = A sin(Bx + C) + D, the midline is y = D.
8. Why is a graph included?
The graph gives a visual check. It helps confirm the range, midline, and wave height, especially when the equation contains shifts or reflections.