The graph-model and residual explorer fits a simple model to a small data set and then shows the residuals, so you can judge how well the model describes the data. It also tells you whether a prediction is an interpolation or an extrapolation.
You can choose a straight line, a quadratic, an exponential or a direct proportion.
How do you use it?
- In Data points, write one point per line as x, y. Use between 3 and 30 points.
- Choose a Model: straight line y = mx + c, quadratic y = ax² + bx + c, exponential y = a × bˣ, or direct proportion y = kx.
- Enter Predict y at x = for the value you want to estimate.
- Press Fit the model. Press Reset to return to the sample.
How do you read the result?
The first line is the fitted model with its parameters. The method is least squares, which picks the parameters that make the sum of squared residuals smallest.
The residual table lists each x, the actual y, the predicted y and the residual. A plus sign means the point is above the model. Below the table are the sum of squared residuals and R².
If all residuals are zero, the tool says the model passes exactly through every point. Otherwise it asks you to read the signs in order of x. A run such as − − + + + − suggests a curve the model missed, while mixed signs suggest a reasonable shape.
The prediction section states whether your x is inside the data range. The graph shows the points, the model and vertical residual lines.
Example walk-through
The sample points are (0, 1), (1, 3), (2, 5), (3, 7), (4, 9) with the straight line model. The fit is y = 2x + 1, every residual is 0 and R² is 1.
The prediction at x = 6 gives y = 13. Because 6 is outside the data range 0 to 4, the tool calls it an extrapolation and warns that it assumes the pattern continues.
Now change the data to (0, 1), (1, 3), (2, 4), (3, 7), (4, 9).
The line becomes y = 2x + 0.8 and the residuals are +0.2, +0.2, −0.8, +0.2, +0.2. They add to 0.
The sum of squared residuals is 0.8 and R² is about 0.98. The point (2, 4) is the one the line misses most.
For a different shape, use (0, 1), (1, 2), (2, 4), (3, 8), (4, 16). The straight line gives residuals of +2, −0.6, −2.2, −1.8 and +2.6, a curved sign pattern. Switch to exponential and the fit is y = 1 × 2^x with residuals at or extremely close to zero.
What are the assumptions and limits?
- Only the four listed models are available. It is not a search through every possible curve.
- A model describes the data you entered. It cannot promise what will happen later.
- Exponential fits need every y to be positive, and quadratics need at least three different x values.
- A good R² alone does not show the model is the right kind.
Which lessons explain the ideas behind it?
- Interpret gradient and intercept in a context explains what m and c mean with units.
- Translate a simple graph and recognise the effect of reflecting a graph connect equations to shapes.
- Compare graphical and exact solutions helps with reading a fitted curve against an exact answer.
- Read a distance-time graph and interpret area under a speed-time graph apply graph reading to motion.
- Choose graph axes from a model links a model to axes in Physics.
- My graph looks correct but uses the wrong scale helps with scale errors.
The wider topic is graphs and transformations, and the mixed practice set tests it.
For a teacher to check your reasoning, see online one-to-one Mathematics tuition or Physics tuition. Other tools are in the learning tools directory.