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Exponential Function Calculator

Enter the two functions and their numeric expression in the calculator, and it will calculate the exponential function, with step-by-step calculations

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Exponential Function Calculator

Use our Exponential Function Calculator to quickly find the exponential function that passes through two points on a coordinate plane. This tool helps you determine if the function is growing or decaying over time.

What is an Exponential Function?

An exponential function is expressed as:

f(t) = A₀ e^(kt)

where:

A₀ is the initial value (at t = 0)
k represents the growth rate (k > 0) or decay rate (k < 0)
t is the independent variable, often time

To determine the exponential function from two points, you need the coordinates (t₁, y₁) and (t₂, y₂).

Finding the Exponential Function

Example:

Suppose we have two points: (t₁, y₁) = (1, 2) and (t₂, y₂) = (4, 8). We want to find the exponential function and compute its value at t = 4.

Solution:

Step 1: Start with the general form:

f(t) = A₀ e^(kt)

Step 2: Set up equations using the given points:

y₁ = A₀ e^(k t₁)

y₂ = A₀ e^(k t₂)

Step 3: Solve for k

Divide the second equation by the first to remove A₀:

y₂ / y₁ = (A₀ e^(k t₂)) / (A₀ e^(k t₁))

y₂ / y₁ = e^(k(t₂ - t₁))

Take the natural logarithm:

ln(y₂ / y₁) = k (t₂ - t₁)

k = ln(y₂ / y₁) / (t₂ - t₁)

Step 4: Solve for A₀

Using one of the points:

A₀ = y₁ e^(-k t₁)

Step 5: Substitute values

k = ln(8 / 2) / (4 - 1) ≈ 0.6931

A₀ = 2 e^(-0.6931 × 1) ≈ 1

Step 6: Final exponential function

f(t) = 1 × e^(0.6931 t) = e^(0.6931 t)

Step 7: Evaluate at t = 4

f(4) = e^(0.6931 × 4) ≈ 8

This function shows exponential growth and passes through the points. A graph can visually confirm the increasing trend.

Graph:

Exponential Function Graph

How the Exponential Function Calculator Works

Input:

Enter the first point (t₁, y₁)
Enter the second point (t₂, y₂)
Optionally, enter a value to evaluate
Click “Calculate”

Output:

The resulting exponential function
Step-by-step calculation details
A graph showing growth or decay

FAQs

What types of exponential functions exist?

Exponential functions can be:

Growing (k > 0)
Decaying (k < 0)

The calculator handles both types accurately.

References

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