Overview
This lecture explains how to linearize non-linear data in AP Physics by transforming graphs so they become linear and easier to analyze.
Linear Relationships
- A linear graph fits the equation y = mx + b, where m is the slope and b is the y-intercept.
- Slope can be calculated using the slope formula, and equations can be found with the point-slope formula.
- Directly proportional relationships appear as a straight line where y increases with x.
Non-Linear Relationships & Linearization
- When data is not linear (e.g. quadratic, inverse, or square root), linearization is used to make it linear.
- Quadratic relationships (parabolas) should be linearized by graphing x² on the x-axis instead of x.
- Inverse relationships (decay) should be linearized by graphing 1/x on the x-axis.
- Square root relationships are linearized by graphing √x or x^(1/2) on the x-axis.
- Sometimes for square root relationships, you can alternatively graph y² versus x.
Graphing & Steps for Linearization
- Identify the type of non-linear relationship from the graph shape (quadratic, inverse, or square root).
- Adjust the x-values according to the relationship: square them, take their reciprocal, or square root them.
- Replot the data with new x-values and draw a new line of best fit.
- The process involves recognizing the pattern and adjusting the x-axis accordingly, as per memorized rules.
Cheat Sheet: Common Relationships & Linearization Rules
- No relationship: y does not change with x (horizontal line), equation y = b, no linearization needed.
- Direct (linear): y = mx + b, no transformation needed.
- Inverse: linearize by graphing y vs. 1/x.
- Quadratic: linearize by graphing y vs. x².
- Square root: linearize by graphing y vs. √x or y² vs. x.
Key Terms & Definitions
- Linearization — the process of transforming a non-linear graph into a linear one for analysis.
- Line of Best Fit — a straight line drawn through data points to represent the general trend.
- Directly Proportional — a relationship where y increases directly as x increases (linear).
- Inverse Proportional — a relationship where y decreases as x increases, following y ∝ 1/x.
- Quadratic Relationship — a relationship where y is proportional to x².
- Square Root Relationship — a relationship where y is proportional to √x.
Action Items / Next Steps
- Memorize the common nonlinear relationships and their required linearization transformations for the AP exam.
- Practice identifying graph shapes and applying the correct transformation.
- Review the provided cheat sheet for quick reference.