Linear Interpolation Calculator

Enter two known data points (x₀, y₀) and (x₁, y₁) and the x-value for which you want to find the corresponding y-value.

What is Linear Interpolation?

Linear Interpolation is a simple method used to estimate an unknown value that lies between two known data points. It works by assuming that the relationship between the points can be reasonably approximated by a straight line.

Imagine you have two points on a graph, (x₀, y₀) and (x₁, y₁). If you want to find the corresponding y value for an x value that sits somewhere between x₀ and x₁, linear interpolation draws a straight line connecting the two known points and finds the y value on that line directly above your target x.

It's a common technique used in many fields, including mathematics, statistics, computer graphics, and engineering, for filling in gaps in data or creating smooth transitions.

How This Linear Interpolation Calculator Works

This calculator uses the standard formula for linear interpolation to find the unknown y value based on the points you provide.

1. Input Known Points: Enter the coordinates of your two known data points:
   • Point 1: (x₀, y₀)
   • Point 2: (x₁, y₁)
2. Input Target x: Enter the specific x-value (x) for which you want to estimate the corresponding y-value.
3. Click Calculate: Press the "Calculate Interpolated y" button.
4. Calculation: The calculator applies the linear interpolation formula: y = y₀ + (x - x₀) * (y₁ - y₀) / (x₁ - x₀) It first calculates the slope of the line between the two points ((y₁ - y₀) / (x₁ - x₀)) and then uses that slope to find the change in y from y₀ based on the change in x from x₀.
5. Validation & Output: The calculator checks if x₀ and x₁ are different (to avoid division by zero). If valid, it displays the calculated y value corresponding to your target x. It also indicates if the target x falls outside the range of x₀ and x₁ (extrapolation).

Frequently Asked Questions (FAQs)

Q1: What's the difference between Interpolation and Extrapolation?

Interpolation involves estimating a value *within* the range of your known data points (i.e., when your target x is between x₀ and x₁).
Extrapolation involves estimating a value *outside* the range of your known data points (i.e., when x is less than both x₀ and x₁, or greater than both).

This calculator uses the same linear formula for both, but extrapolation is generally considered less reliable because you are assuming the linear trend continues beyond your observed data, which may not be true.

Q2: When is Linear Interpolation most useful?

Linear interpolation is best suited for situations where:

• You have data points that are relatively close together.
• You can reasonably assume the relationship between the points is close to linear (doesn't curve significantly between the points).
• You need a quick and simple way to estimate missing values.

It's commonly used for tasks like reading values between entries in a table, simple data smoothing, or basic animation/graphics calculations.

Q3: What are the limitations of Linear Interpolation?

The main limitation is its assumption of linearity. If the actual relationship between the data points is curved (e.g., exponential, logarithmic, or polynomial), linear interpolation will produce an estimate that deviates from the true value. The further apart the known points are, or the more curved the actual function is, the less accurate linear interpolation will be. For more complex relationships, methods like polynomial interpolation or spline interpolation might be necessary.

Q4: Can I input negative numbers or decimals?

Yes, the calculator accepts positive numbers, negative numbers, and decimals for all input values (x₀, y₀, x₁, y₁, x).

Q5: What happens if x₀ = x₁?

If the two known x-values (x₀ and x₁) are the same, the formula involves division by zero (x₁ - x₀ = 0), which is mathematically undefined. The calculator will detect this and display an error message, as interpolation requires two distinct points along the x-axis.

Q6: Does the order of Point 1 and Point 2 matter?

No, the order in which you enter the two known points (x₀, y₀) and (x₁, y₁) does not affect the final result. The formula calculates the slope and position correctly regardless of which point is designated as "Point 1" or "Point 2". However, the calculator does use the designation to check if the target `x` is between `x0` and `x1` for the extrapolation warning.