Curve Sketching

AP Calculus · Axiom Academy

Combining derivatives to create a complete picture of function behavior The Complete Curve Sketching Process Curve sketching brings together everything we've learned about derivatives into a single, powerful technique. By systematically analyzing a function, we can sketch an accurate graph without a calculator. Find intercepts ( x -intercepts where f(x)=0 , y -intercept at f(0) ) Find asymptotes (vertical, horizontal, oblique) Find critical points: f'(x) = 0 or undefined Find intervals of increase/decrease using f'(x) Find concavity and inflection points using f''(x) Intercepts: y -intercept at (0, 0) ; x -intercept at (0, 0) Symmetry: f(-x) = f(x) — the function is even Asymptotes: Vertical at x = 1 and x = -1 ; Horizontal at y = 1 (since degree numerator = degree denominator) f'(x) = 0 at x = 0 . Sign analysis: f'(x) > 0 for x 0 (decreasing). So x = 0 is a local maximum. Second derivative analysis confirms concavity changes near the asymptotes. Curve sketching is the synthesis of all derivative analysis Always check domain and asymptotes first — they define the skeleton f'(x) reveals increasing/decreasing and extrema f''(x) reveals concavity and inflection points Symmetry and intercepts fill in the details

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