We fix a flexible model — a degree-8 polynomial fit with ridge regularization — and use the regularization strength λ as the single knob for model complexity. A large λ shrinks the coefficients hard, giving a stiff fit that is the same no matter which data we see (low variance) but systematically off (high bias). A small λ lets the curve chase every wiggle, so it is nearly unbiased but swings wildly from sample to sample (high variance). To see this, we draw B = 40 bootstrap resamples of the data, refit at each λ, and watch how the fitted curves spread out.
The dashed green curve is a stand-in for the unknown true function f(x): it is just a lightly-regularized fit on the full dataset (smallest λ). This is only an approximation of the truth — we never know f exactly — but it lets us decompose error into bias and variance. The irreducible noise σ2 is estimated as the residual variance of that reference fit.