Exponential functions play a significant role in various fields such as finance, economics, and sciences. Being able to determine an exponential function given two points not only helps us understand patterns and trends in data but also enables us to make predictions and projections for the future. In this article, we will explore the step-by-step process of determining an exponential function using two given points. Let’s dive in!

What is an exponential function?

An exponential function is a mathematical function in which the independent variable appears as an exponent. It is written in the form of f(x) = a * b^x, where ‘a’ is a constant, ‘b’ is the base, and ‘x’ represents the independent variable.

Why is it important to determine an exponential function?

Determining an exponential function allows us to model real-life phenomena that exhibit exponential growth or decay. This knowledge is valuable for making predictions, understanding trends, and forming informed decisions in various fields.

3. How to determine an exponential function given two points:
Step 1: Identify the two given points (x₁, y₁) and (x₂, y₂) in the dataset.
Step 2: Substitute the coordinates of each point into the exponential function form, f(x) = a * b^x, to get two equations.
– For the first point, x₁ and y₁, we have: y₁ = a * b^(x₁)
– For the second point, x₂ and y₂, we have: y₂ = a * b^(x₂)

4. How to find ‘a’ and ‘b’ in the exponential function:
Step 3: Divide the second equation by the first equation to eliminate ‘a’:
– y₂ / y₁ = (a * b^(x₂)) / (a * b^(x₁))
– Simplifying gives us: y₂ / y₁ = b^(x₂ – x₁)

Step 4: Solve for ‘b’ by isolating it:
– Taking the logarithm of both sides gives: log(b) = (x₂ – x₁) * log(y₂ / y₁)
– Therefore, we have: b = 10 ^ ((x₂ – x₁) * log(y₂ / y₁))

Step 5: Substitute the calculated value of ‘b’ back into any of the original equations (either y₁ = a * b^(x₁) or y₂ = a * b^(x₂)) to solve for ‘a’.
– Rearranging the equation gives us: a = y₁ / (b^(x₁))

5. Example:
Let’s consider two points, (1, 3) and (2, 12), to determine the exponential function equation.

Step 1: Given points: (1, 3) and (2, 12)
Step 2: Equations: 3 = a * b^(1) and 12 = a * b^(2)

Step 3: Dividing the second equation by the first equation gives: 12 / 3 = b^(2 – 1), resulting in 4 = b^1.

Step 4: Solving for ‘b’ by taking the logarithm of both sides: log(b) = log(4), which gives us b = 10^(log(4)). Therefore, b = 10^0.6021 ≈ 4. And we have found the value of ‘b.’

Step 5: Substituting ‘b’ into the first equation, we find: 3 = a * 4^(1). Solving for ‘a’ gives us a = 3 / 4, which is approximately 0.75.

Therefore, the exponential function equation passing through the given points is: f(x) ≈ 0.75 * 4^x.

Determining an exponential function with the help of two given points is a powerful tool for understanding patterns and making predictions. By following the step-by-step process outlined in this article, you can easily find the values of ‘a’ and ‘b’ in the exponential function equation. Remember, exponential functions showcase valuable trends and patterns, enriching our ability to analyze and project various real-life scenarios.

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