A research plan for sketching a graph
Study a function by a plan: domain, parity, zeros, increasing and decreasing, extreme points, then mark the key points of the graph.
- Germany: grade 11
- Ukraine: grade 10
- Russia: grade 11
Guide
This follows the Ukrainian Grade 10 course (algebra and calculus, section 3, item 25, Graph sketching). A function is studied by a plan: 1) domain; 2) parity; 3) zeros; 4) intervals of increase and decrease; 5) extreme points; 6) other features. The graph is built from these data.
Example from the textbook: f(x) = (32)x2 − (14)x3. Domain: D(f) = ℝ. The function is neither even nor odd. Zeros: x2(32 − x4) = 0, so x = 0 and x = 6. Derivative: f′(x) = (3x4)(4 − x); the sign of f′ changes from − to + at 0 and from + to − at 4. So xmin = 0, xmax = 4, and f(4) = 8, f(0) = 0.
An even function has a graph symmetric about the y-axis, an odd one about the origin, so it is enough to study the function for x ⩾ 0.