Formula Used
The calculator uses the shifted exponential model:
y = k + a × b(x - h)
Here, a is the coefficient, b is the base, h is the horizontal shift, and k is the vertical shift.
For two points, the adjusted values are Y1 = y1 - k and Y2 = y2 - k. Then:
b = (Y2 / Y1)1 / (x2 - x1)
a = Y1 / b(x1 - h)
For a percent growth rate, b = 1 + r / 100. For decay, b = 1 - r / 100.
How to Use This Calculator
- Select the method that matches your available information.
- Enter two points, a starting value with rate, or a coefficient with base.
- Add optional horizontal and vertical shifts when needed.
- Set the x-range for the generated table.
- Press the submit button to show the equation above the form.
- Use the CSV or PDF button to save the result.
Example Data Table
| Method |
Inputs |
Expected Equation |
Meaning |
| Two points |
(0, 3), (2, 12) |
y = 3 × 2x |
The value doubles each unit. |
| Starting value and growth |
a = 100, r = 8% |
y = 100 × 1.08x |
The value grows by eight percent. |
| Starting value and decay |
a = 80, r = 15% |
y = 80 × 0.85x |
The value falls by fifteen percent. |
Understanding Exponential Equations
An exponential equation describes repeated multiplication. It is useful when a quantity changes by a constant factor over equal steps. The model appears in finance, biology, chemistry, physics, population study, and algebra. A simple form is y = a × b raised to x. The coefficient gives the starting size. The base shows the multiplier. A base greater than one means growth. A base between zero and one means decay.
Why This Calculator Helps
Writing the equation by hand can be slow. You may know two points but not the base. You may know a rate but not the multiplier. You may also need shifts for a transformed graph. This calculator handles these cases in one place. It adjusts the values, finds the base, builds the model, and creates a table for checking.
Using Two Points
Two points are enough when the values are positive after any vertical shift. The calculator divides the adjusted y-values. It then uses the distance between the x-values. This gives the base. After that, it solves for the coefficient. The result is an equation that passes through both entered points.
Growth and Decay Rates
Percent rates are common in real examples. A ten percent increase uses a base of 1.10. A ten percent decrease uses a base of 0.90. This calculator converts the percent into a base automatically. It also labels the model as growth or decay.
Shifts and Tables
Advanced equations may include horizontal and vertical shifts. The horizontal shift changes the exponent. The vertical shift moves the graph up or down. The value table helps you inspect the pattern. It is also useful for lessons, homework, reports, and quick comparisons. Export options make the answer easier to reuse.
FAQs
What does this calculator write?
It writes an exponential equation from two points, a starting value with rate, or a known coefficient and base.
What is the standard exponential form?
The common form is y = a × b^x. This calculator also supports shifted form y = k + a × b^(x - h).
What does the base mean?
The base is the repeated multiplier. If it is greater than one, the model grows. If it is between zero and one, the model decays.
Can I use negative y-values?
For the two-point method, y minus the vertical shift must be positive. This keeps the base calculation real and usable.
How is percent growth handled?
The calculator changes percent growth into a multiplier using b = 1 + r / 100.
How is percent decay handled?
The calculator changes percent decay into a multiplier using b = 1 - r / 100.
Why add a value table?
The table shows predicted y-values across your chosen x-range. It helps verify the pattern and explain the equation.
Can I export my result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a clean printable summary.