Function Graphs, Explained Visually — Symmetry, Shifts & Extrema

You've seen functions as rules. This animated lesson is about seeing them — the graph: how to read it, how to build one point by point, the symmetry you can spot at a glance, the vertical line test, intercepts, the four transformations that reshape any graph, and how to describe where a function rises, falls, and peaks. No prior calculus needed. If you're heading into calculus or precalculus — or want a solid refresher — this locks in the visual side of functions. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ WHAT YOU'LL LEARN ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ • The precise definition of the graph of f: all points (x, f(x)) • The rectangular coordinate system — quadrants, signs, reading off a point • Building a graph honestly: compute a point, verify it, repeat, connect • The reciprocal function y = 1/x: its domain and asymptotic shape • Odd symmetry, f(−x) = −f(x) (about the origin), and even symmetry, f(−x) = f(x) (about the y-axis) • The vertical line test — and exactly why a sideways parabola fails it • x-intercepts and the y-intercept, with a worked example • The four transformations: vertical & horizontal shifts, stretches & compressions, reflections — and the correct order to apply them • A flagship worked example: building f(x) = (x+1)/(x−2) from g(x) = 1/x, step by step • Increasing, decreasing, and constant functions • Extrema: local vs. absolute maximum and minimum, stated precisely • A short self-check quiz — pause and try each one ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ CHAPTERS ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 0:00 What the graph of a function is 0:24 The coordinate system & plotting points 1:04 Building a graph from points 2:24 The reciprocal function y = 1/x 3:15 Odd functions (symmetry about the origin) 4:12 Even functions (symmetry about the y-axis) 5:09 The vertical line test 6:07 Intercepts 6:48 Worked example: intercepts of x^2 - 2x 7:29 Shifting graphs 9:39 Stretching & compressing graphs 11:34 Reflecting graphs 12:50 Worked example: building (x+1)/(x-2) from 1/x 14:08 Increasing & decreasing 14:36 Constant functions 15:06 Extrema: the big picture 16:02 Local maximum & minimum 16:40 Absolute maximum & minimum 17:23 Recap 18:20 Check yourself (quiz) ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ ABOUT ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Part of the Applied Calculus series. Animations made with Manim. Follows the "Functions" lesson; sets up everything you need before calculus proper. If this helped, a like and a subscribe genuinely help the channel — thank you. #calculus #precalculus #functions #graphs #graphingfunctions #transformations #symmetry #mathhelp #stem Keywords: graph of a function, graphing functions, coordinate system, quadrants, plotting points, reciprocal function, 1/x graph, asymptote, odd function, even function, symmetry, vertical line test, x-intercept, y-intercept, transformations of functions, vertical shift, horizontal shift, stretch and compress, reflection, order of transformations, increasing and decreasing, constant function, local maximum, local minimum, absolute maximum, absolute minimum, extrema, calculus, precalculus, calculus foundations, animated math, Manim