# Chapter 12: Sketching Graphs > Learn the sign-table craft: read increasing/decreasing from f′ and concavity from f″, then assemble any function into a reliable sketch using the Big Method. **Canonical:** https://calculus.kitzo.work/chapter/sketching-graphs **Level:** Core **Estimated time:** 40 minutes **Exercises:** 8 (auto-marked; answers stay server-side) ## Topics - Table of signs for f′ - Table of signs for f″ - The Big Method - Worked example: polynomial - Worked example: rational function - Asymptotes & intercepts **Why this matters** A sketch is a claim your derivative makes about the world. Sign tables turn f′ and f″ from formulas into a map of the curve: where it rises, where it falls, where it cups, where it caps, and where it settles or escapes. Master this and you will never need to plot twenty points again — two small tables will build the picture for you. ## 1 · Why sign tables matter A derivative is a rate. Its **sign** — not its size — tells you the direction of travel. f′(x) > 0 means the function is increasing; f′(x) < 0 means decreasing. Likewise f″ tells the direction of bending: f″(x) > 0 concave up, f″(x) < 0 concave down. A sign table condenses this into one glance: split the number line at every critical point (zeros of f′, zeros of f″, or points where either fails to exist), then test one x per interval. ## 2 · Table of signs for f′ Recipe: find where f′ vanishes or is undefined, sort them, then test one x per interval and record + or −. Between consecutive sign changes the function is monotonic; a + → − change marks a local max, − → + a local min, and no change means the derivative never actually switched direction. **Worked example 1** ### Sign table for f′ of f(x) = x³ − 3x - **Differentiate:** f′(x) = 3x² − 3 = 3(x − 1)(x + 1). Zeros at x = −1 and x = 1. - **Test intervals:** x = −2 gives f′ = 9 > 0; x = 0 gives f′ = −3 < 0; x = 2 gives f′ = 9 > 0. - **Read the shape:** increasing on (−∞, −1), decreasing on (−1, 1), increasing on (1, ∞). So x = −1 is a local max (value 2) and x = 1 is a local min (value −2). ## 3 · Table of signs for f″ Do the same with f″. Its zeros split the line into intervals of fixed concavity. Where f″ changes from + to − (or back), the curve switches from cupping to capping — an inflection point. For our cubic, f″(x) = 6x, which is negative for x < 0 (concave down) and positive for x > 0 (concave up). The sign changes at x = 0, so (0, 0) is the inflection point, exactly halfway between the max and the min. ## 4 · The Big Method The Big Method is the full pipeline for any curve. Work through the steps in order — each one removes a whole class of possible surprises: - **Domain:** where is f defined? Note holes, vertical asymptotes, and endpoints. - **Intercepts:** set x = 0 and y = 0; find where the curve crosses the axes. - **Asymptotes:** vertical at denominators that vanish (with nonzero numerator); horizontal from leading-end behaviour — compare degrees, or take the limit at ±∞; oblique when the numerator's degree exceeds the denominator's by one. - **Sign table for f′:** intercepts of f′ give extrema and monotonicity. - **Sign table for f″:** concavity and inflection points. - **Assemble:** plot intercepts, extrema, inflection points, and asymptotes; connect them smoothly; verify against a grapher. **Worked example 2** ### Sketch f(x) = x³ − 3x with the Big Method - **Domain:** all reals (polynomial). - **Intercepts:** f(0) = 0; y = 0 gives x(x² − 3) = 0 → x = 0, ±√3 ≈ ±1.73. - **Asymptotes:** none (polynomial), but degree 3 with odd leading power: escapes to −∞ left, +∞ right. - **f′ table:** from Example 1 — increasing (−∞, −1), decreasing (−1, 1), increasing (1, ∞). Local max (−1, 2), local min (1, −2). - **f″ table:** f″ = 6x; concave down on (−∞, 0), concave up on (0, ∞). Inflection at (0, 0). - **Assemble:** draw the curve entering from the bottom-left, rising through (−√3, 0) to (−1, 2), falling through (0, 0) to (1, −2), rising through (√3, 0) and exiting top-right. One inflection at the origin, symmetric about it (the cubic is odd). ## 5 · Rational functions and their asymptotes For rational functions the asymptotes carry the sketch. f(x) = 1/(x² + 1): domain all reals (x² + 1 never vanishes, so no vertical asymptotes); f(0) = 1 maximum; y = 0 is a horizontal asymptote at both ends; f′(x) = −2x/(x² + 1)² vanishes only at x = 0 (the peak); f″ changes sign at x = ±1/√3 ≈ ±0.58 — two inflection points flanking the peak. The result is a bell curve, every point below y = 1, hugging the x-axis as |x| → ∞. Contrast with g(x) = x/(x + 1)²: the denominator vanishes at x = −1 with nonzero numerator, so x = −1 is a **vertical asymptote**; the denominator outgrows the numerator, so y = 0 is the **horizontal asymptote**. Intercepts: x = 0 (value 0) is the local max; the curve plunges to −∞ as x → −1 from the left and rebounds from +∞ on the right, then settles toward 0 from above as x → ∞. Locate intercepts, asymptote behaviour, and extrema before drawing a single stroke — asymptotes are scaffolding, not decoration. **Watch out** A zero denominator does not automatically mean a vertical asymptote — cancel common factors first. Likewise, if f′(x) = 0 at a point but the sign does not change, it is not an extremum. And infinity behaviour must be checked at both ends: a function can hug y = 0 on the left and blow up on the right if the degrees only mismatch at one side (or if your domain is one-sided). ## 6 · Verify with a grapher A grapher is the umpire, not the coach. Sketch first, then plot and check every feature — intercepts, extrema, inflection points, asymptotes — appears where you predicted. Any mismatch means a sign-table slip or a missed asymptote. **Chapter formula sheet** f′ > 0 ⇔ f increasing on that interval f′ < 0 ⇔ f decreasing on that interval f′ changes + → − ⇒ local max; − → + ⇒ local min f″ > 0 concave up; f″ < 0 concave down Inflection point ⇔ f″ changes sign Vertical asymptote: denominator 0, numerator ≠ 0 Horizontal asymptote: lim x→±∞ f(x) ## Common mistakes - Sketching before computing f′ and f″ — drawing by eye guarantees missed extrema and wrong concavity. - Missing the sign-change check at a zero of f″ and inventing inflection points. - Forgetting vertical asymptotes can occur only where the denominator vanishes after cancelling. - Assigning a horizontal asymptote by looking at one end only. - Letting the curve cross a vertical asymptote — it never does; behaviour on each side must be checked separately. - Comparing a sketch to a grapher without asking *why* each feature appears. ## Practice exercises 1. On the interval (1, 4), f′(x) is negative. What does this tell you about f on that interval? - A. f is increasing on (1, 4) - B. f is decreasing on (1, 4) - C. f is concave up on (1, 4) - D. f has an inflection point at x = 2.5 2. Find the local maximum value of y = x³ − 3x. 3. State the horizontal asymptote of y = 1/(x² + 1). Answer as e.g. y=0. 4. Which vertical asymptote does the curve y = x/(x + 1)² have? - A. x = 0 - B. x = 1 - C. x = −1 - D. no vertical asymptote 5. Give the x-coordinate of the inflection point of y = x³ − 3x. 6. Find the local minimum value of y = x³ − 3x. 7. f″(x) > 0 on an interval. The graph of f is: - A. concave down there - B. concave up there - C. decreasing there - D. flat there 8. Find the value c such that y = (3x² + 1)/(x² + 5) has horizontal asymptote y = c. --- Practice interactively: https://calculus.kitzo.work/chapter/sketching-graphs Citation: CalculusKitzo (https://calculus.kitzo.work/). Independent study notes inspired by the standard calculus syllabus; not affiliated with Princeton University Press. See https://calculus.kitzo.work/disclaimer