Chapter 2: Functions and Relations
A mathematician, like a painter or a poet, is a maker of patterns.
Why This Matters
Functions are the verbs of mathematics. They describe how things change — how one quantity depends on another.
When you understand functions, you understand:
-
How a ball falls: \(h(t) = -16t^2 + v_0t + h_0\)
-
How populations grow: \(P(t) = P_0 e^{rt}\)
-
How networks scale: \(f(n) = \log n\)
Sections
| Section | Topic | Key Skill | Status |
|---|---|---|---|
The Rectangular Coordinate System |
Plot, distance, midpoint |
[ ] Not Started |
|
Circles |
Standard form \((x-h)^2 + (y-k)^2 = r^2\) |
[ ] Not Started |
|
Functions and Relations |
Domain, range, function notation |
[ ] Not Started |
|
Linear Equations in Two Variables |
Slope, intercepts |
[ ] Not Started |
|
Applications of Linear Equations |
Modeling |
[ ] Not Started |
|
Transformations of Graphs |
Shifts, reflections, stretches |
[ ] Not Started |
|
Analyzing Graphs |
Symmetry, increasing/decreasing |
[ ] Not Started |
|
Algebra of Functions |
\((f+g)(x)\), \((f \circ g)(x)\) |
[ ] Not Started |
Mathematical Wisdom
The function is perhaps the most important concept in modern mathematics.
Consider \(f(x) = x^2\). This simple rule takes any number and squares it. But what makes it profound is the mapping — each input has exactly one output.
When you compose functions, \(f(g(x))\), you’re chaining transformations. Input flows through \(g\), then through \(f\). This idea underlies everything from cryptography to machine learning.
The graph of a function is a visual story:
-
Rising: The function is increasing
-
Falling: The function is decreasing
-
Flat: The function is constant
Functions don’t just compute. They reveal structure.
Study Tip
Sketch graphs by hand before reaching for a calculator. The act of transforming \(y = x^2\) into \(y = (x-2)^2 + 3\) by shifting builds geometric intuition that no calculator can provide.