Functions, Graphs & Coordinate Geometry

Overview

Before any AI math, get comfortable with the single most important object in the field: the function. f(x) = 2x + 1 takes an input and returns an output; its graph is a straight line whose slope (2) and intercept (1) you can read off directly. This matters because a machine-learning model IS a function — a huge one with millions of tunable numbers — and "training" means adjusting those numbers so the function's graph passes through your data. Slope and intercept reappear as "weight" and "bias", the two quantities every linear layer learns. Coordinate geometry (points, distance, lines in 2D) is the ground on which vectors and higher-dimensional spaces are built, so a firm picture here pays off everywhere later.

A line: slope and intercept = weight and bias

import numpy as np

w, b = 2.0, 1.0                 # weight (slope), bias (intercept)
x = np.linspace(-3, 3, 7)       # inputs: -3, -2, ... 3
y = w * x + b                   # the line / a linear neuron
print(y)   # [-5. -3. -1.  1.  3.  5.  7.]

# Change w -> steeper line; change b -> whole line moves up/down.
# Training a model = searching for the w and b that fit the data.

Non-linear functions and why AI needs them

import numpy as np

x = np.linspace(-6, 6, 5)
relu    = np.maximum(0, x)          # bends at 0: flat then rising
sigmoid = 1 / (1 + np.exp(-x))      # smooth S-curve squashing to (0,1)

print("relu   ", relu)      # [0. 0. 0. 3. 6.]
print("sigmoid", np.round(sigmoid, 3))  # [0.002 0.047 0.5 0.953 0.998]

Key Points to Remember

• 1A function maps inputs to outputs; its graph is that mapping drawn out
• 2y = wx + b: slope=weight, intercept=bias — the core of a linear layer
• 3Non-linear functions (ReLU, sigmoid) let networks fit curved data
• 4Reading a graph = predicting a function's behaviour without computing every point