1. What is Linear Interpolation?

Definition: Linear interpolation is a method of estimating values between two known values in a sequence of points.

Purpose: It's commonly used in mathematics, computer graphics, data analysis, and engineering to estimate intermediate values.

2. How Does Linear Interpolation Work?

The calculator uses the formula:

Where:

• \( y \) — Interpolated value we want to find
• \( y₁, y₂ \) — Known y-values at points x₁ and x₂
• \( x \) — Point where we want to estimate the value
• \( x₁, x₂ \) — Known x-values (x₁ ≠ x₂)

Explanation: The formula calculates the slope between two known points and applies this slope to estimate the value at the desired point.

3. Applications of Linear Interpolation

Details: Used in:

• Estimating values in tables or datasets
• Computer graphics for smooth transitions
• Engineering calculations between known data points
• Financial modeling and forecasting

4. Using the Calculator

Tips: Enter two known (x,y) points and the x-value where you want to estimate y. Ensure x₁ and x₂ are different values.

5. Frequently Asked Questions (FAQ)

Q1: What if my x-value is outside the known range?
A: This is called extrapolation. While possible, it's less accurate than interpolation between points.

Q2: Why do I get an error when x₁ = x₂?
A: The formula requires division by (x₂ - x₁), so these values must be different to avoid division by zero.

Q3: How accurate is linear interpolation?
A: It's exact for linear relationships and a good approximation for nearly-linear data between close points.

Q4: Can I use this for non-numeric data?
A: No, linear interpolation only works with numeric values that can be placed on a continuous scale.

Q5: What's the difference between interpolation and regression?
A: Interpolation fits exactly through known points, while regression finds the best-fit line that may not pass exactly through any point.