These vertical distances are called residuals. Linear regression finds the line that minimizes the sum of squared residuals -- this is called Ordinary Least Squares (OLS). The line is $y = \beta_0 + \beta_1 x$.

The line is the OLS fit: $y = \beta_0 + \beta_1 x$. Points close to the line suggest a strong linear relationship. Predictions near the center of the data are more reliable than extrapolations. The $R^2$ value measures how much variance the line explains.

A good residual plot shows random scatter around zero -- no patterns. If you see curves or funnels, the model is missing something (non-linearity or heteroscedasticity). A "perfect" plot would be a flat band of random points.

This is gradient descent -- at each step, move downhill in the steepest direction. Large steps (high learning rate) may overshoot the minimum. Small steps (low learning rate) converge slowly. You stop when steps become negligibly small (convergence).

The wiggly model overfits -- it memorizes noise (high variance). The flat model underfits -- it misses the real pattern (high bias). The best model balances both. This is the bias-variance tradeoff, a fundamental concept in ML.

With multiple features, we use multiple linear regression: $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \ldots$ Each coefficient captures one feature's effect while holding others constant. When features are correlated (e.g., sqft and bedrooms), we face multicollinearity, which makes individual coefficients unreliable.