What is the primary difference between interpolation and extrapolation?

Interpolation estimates values within the range of known data points, while extrapolation estimates values outside that range. Interpolation is generally more reliable because it is constrained by known boundaries on both sides.

How does linear interpolation differ from linear regression?

Linear interpolation creates a line that must pass exactly through two specific data points to estimate intermediate values. Linear regression creates a 'best-fit' line for a whole dataset that minimizes overall error but may not pass through any individual point.

When should you choose cubic interpolation over linear interpolation?

Cubic interpolation should be used when the data is known to be smooth and curved rather than composed of straight segments. It provides a more accurate fit for non-linear functions but requires more data points and computational power.

What is the 'Boundary Error' in linear interpolation?

This error occurs when a calculated value falls outside the range of the two known $y$-values. It usually indicates a calculation mistake, such as swapping $x$ and $y$ or using the wrong interval points.

Why is linear interpolation risky for data with high curvature?

Because it assumes a straight-line path, it will significantly over-estimate or under-estimate values if the real function curves sharply between the points. The straight line 'cuts the corner' of the actual curve.

What happens if you use the wrong interval (e.g., $x$ is not between $x_1$ and $x_2$)?

You are performing extrapolation instead of interpolation. The linear trend between the chosen points may not apply to the region where your target $x$ resides, leading to highly inaccurate results.

Define the 'Interpolation Fraction' ($t$).

The interpolation fraction is the ratio $\frac{x - x_1}{x_2 - x_1}$, representing how far the target $x$ is through the interval. It ranges from 0 (at $x_1$) to 1 (at $x_2$).

What is the point-slope formula used in linear interpolation?

The formula is $y = y_1 + (x - x_1) \frac{y_2 - y_1}{x_2 - x_1}$. It calculates the change in $y$ by multiplying the distance from the start ($x - x_1$) by the constant slope.

In the context of computer science, what does 'Lerp' stand for?

Lerp is a shorthand for Linear Interpolation. It is a function that returns a value linearly blended between two endpoints based on a weight parameter, usually between 0 and 1.

What is a 'Data Point' in the context of interpolation?

A data point is a known coordinate pair $(x, y)$ from a set of observations or a table. Interpolation uses these fixed anchors to estimate the values of the 'empty spaces' between them.

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Linear Interpolation

Summary

Linear interpolation is a mathematical method used to estimate unknown values that fall between two known data points by assuming a constant rate of change. By constructing a straight line between these coordinates, one can derive a functional relationship that approximates intermediate values with high efficiency, provided the underlying data behaves linearly.

1. Definition & Core Concepts

A coordinate graph showing a red line connecting two points (x1, y1) and (x2, y2). An intermediate point (x, y) is highlighted in orange on the line, illustrating how its value is derived from the linear relationship.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

Feature	Interpolation	Extrapolation
Location	Inside the range of known points	Outside the range of known points
Reliability	Generally high for smooth functions	Low; assumes trends continue indefinitely
Risk	Small errors due to local curvature	Large errors due to unknown future behavior

Interpolation vs. Regression: While interpolation forces the line to pass exactly through the known data points, linear regression finds a line of best fit that may not pass through any specific point but minimizes overall error across a large dataset.
Linear vs. Polynomial Interpolation: Linear interpolation uses a degree-1 polynomial (a line), whereas higher-order methods (like quadratic or cubic splines) use curves to better fit data that has significant acceleration or change in direction.

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions

Ignoring Curvature: A common mistake is using linear interpolation for data that is highly non-linear (like exponential growth) over a wide interval. This leads to an 'under-estimation' in convex curves and 'over-estimation' in concave curves.
Variable Swapping: Students often accidentally swap the $x$ and $y$ values in the formula. Remember that $x$ is the value you know (the input), and $y$ is the value you are estimating (the output).
Extrapolation Danger: Applying the linear formula to values outside the known range is technically extrapolation. This is dangerous because the linear trend observed between two points may not exist at all beyond those boundaries.