A function is a rule that assigns exactly one output to each input, written \(y = f(x)\) and read \(f\) of \(x\). The domain is the set of valid inputs and the range is the set of actual outputs produced. The vertical line test identifies graphs that represent functions, and the horizontal line test identifies which functions are one-to-one and therefore invertible. Common families include linear functions \(f(x) = mx + b\), quadratic functions \(f(x) = ax^2 + bx + c\), exponential functions, logarithmic functions, polynomial functions, and rational functions. Each family has its own characteristic shape, asymptotes, and growth behavior.
Linear functions describe any constant rate of change. The slope-intercept form \(y = mx + b\) makes the slope \(m\) (rise over run) and y-intercept \(b\) immediately visible; the point-slope form \(y - y_1 = m(x - x_1)\) is convenient when you know one point and the slope. Two lines are parallel when their slopes are equal and perpendicular when their slopes are negative reciprocals. Quadratic functions trace parabolas whose vertex is at \((-b/(2a), f(-b/(2a)))\), the axis of symmetry \(x = -b/(2a)\) being a vertical line through the vertex. Functions can be shifted horizontally or vertically, reflected across the axes, or stretched/compressed by multiplying the input or output by constants.
The Cartesian coordinate system anchors geometry to algebra by representing points as ordered pairs \((x, y)\) on perpendicular axes. The four quadrants correspond to the four sign combinations of the coordinates. The distance formula \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\) extends the Pythagorean theorem to any two points, and the midpoint formula averages each coordinate. Three-dimensional coordinates add a third axis \(z\), with distances computed by summing the squared differences across all three dimensions.
Vectors encode both magnitude and direction, and they support operations tailored to those two aspects. Geometrically, a 2D vector is an arrow; algebraically, it is a pair of components \((v_x, v_y)\). Addition places vectors head-to-tail, scalar multiplication stretches or flips them, and the dot product \(\mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y\) equals \(\|\mathbf{a}\|\|\mathbf{b}\|\cos\theta\), encoding both magnitude and the angle between them. Two vectors are orthogonal precisely when their dot product is zero. The dot product therefore measures directional similarity and underlies everything from projections to cosine similarity to neural attention. The cross product, defined in three dimensions, produces a vector perpendicular to both inputs with magnitude equal to the area of the parallelogram they span.