A derivative-based curve sketching checklist (the “big picture” workflow)
The goal is to sketch an accurate graph with minimal point-plotting by extracting shape information from algebraic features and derivative-based features. You will still evaluate the function at a few carefully chosen x-values to (a) choose a reasonable viewing window and (b) sanity-check the sketch.
Checklist (use in this order)
- Domain: where the formula makes sense (watch denominators, even roots, logs).
- Intercepts: x-intercepts (solve f(x)=0) and y-intercept (f(0) if defined).
- Symmetry (when helpful): even/odd tests, or symmetry about a vertical line (e.g., by rewriting).
- Asymptotes (as needed): vertical (domain breaks with blow-up), horizontal/oblique (end behavior).
- Critical points: where f′(x)=0 or undefined (inside the domain).
- Increasing/decreasing: sign of f′ on intervals.
- Concavity: sign of f″ on intervals.
- Inflection points: where concavity changes (and f is continuous there).
Important: the checklist is not a list of separate tasks; it is a story about how the curve must move. Your sketch should be the simplest curve that satisfies all features simultaneously.
Organize the work: a two-column analysis table
To avoid losing track of details, write everything in a two-column table:
| Feature | Evidence / computation / conclusion |
|---|---|
| Domain | … |
| Intercepts | … |
| Symmetry | … |
| Asymptotes | … |
| Critical points | … |
| Increasing/decreasing | … |
| Concavity | … |
| Inflection points | … |
| Key function values (plausibility) | Pick 2–5 x-values per “region” to confirm heights |
This format forces you to justify each visual feature with a computation and makes it easy to translate analysis into a clean sketch.
From analysis to a clean sketch
Step-by-step sketching procedure
- Step 1: Draw axes and mark asymptotes lightly. Use dashed lines for asymptotes. If there are multiple vertical asymptotes, mark each.
- Step 2: Plot guaranteed points. Intercepts, endpoints of the domain (if any), and any points you computed exactly (e.g., local extrema coordinates).
- Step 3: Mark critical x-values and inflection x-values on the x-axis. These split the domain into intervals where the behavior is uniform (increasing/decreasing, concave up/down).
- Step 4: On each interval, draw the simplest curve consistent with (a) monotonicity and (b) concavity. Make sure the curve approaches asymptotes in the correct direction (up/down) and matches end behavior.
- Step 5: Label intervals and key points. Write “inc/dec” and “CU/CD” (concave up/down) above the x-axis, and label local maxima/minima and inflection points.
- Step 6: Choose/adjust a viewing window. Use a few function values to ensure the graph is not “squashed” or missing important features.
- Step 7: Plausibility check. Evaluate f(x) at a few x-values in each region to confirm the curve’s height and sign.
Interpret the sketch back into words
A complete curve sketch is not finished until you can describe it. Your written description should include:
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- Where the function is defined and any breaks in the graph.
- Where it crosses axes and whether it stays above/below the x-axis on intervals.
- Where it increases/decreases and where it has local extrema.
- Where it is concave up/down and where it changes concavity.
- How it behaves near asymptotes and as x→±∞.
This “graph-to-words” step is a powerful self-check: if you cannot describe a feature clearly, it may not be justified by your analysis.
Guided Example 1: Rational function with two vertical asymptotes
Function: f(x) = (x^2 - 1)/(x^2 - 4)
Two-column analysis table
| Feature | Evidence / computation / conclusion |
|---|---|
| Domain | Denominator ≠ 0 ⇒ x^2-4≠0 ⇒ x≠±2. Domain: (-∞,-2)∪(-2,2)∪(2,∞). |
| Intercepts | x-intercepts: numerator 0 ⇒ x^2-1=0 ⇒ x=±1 (allowed). y-intercept: f(0)=(-1)/(-4)=1/4. |
| Symmetry | Even function: depends on x^2 only ⇒ f(-x)=f(x). Sketch is symmetric about y-axis. |
| Asymptotes | Vertical: x=±2. Horizontal: degrees equal ⇒ ratio of leading coefficients 1/1 ⇒ y=1. |
| Critical points | f(x)=1+3/(x^2-4). Then f′(x)=3·d/dx[(x^2-4)^{-1}] = 3·(-1)(x^2-4)^{-2}·2x = -6x/(x^2-4)^2. Set f′=0 ⇒ x=0 (in domain). f′ undefined at ±2 but those are not in domain. |
| Increasing/decreasing | Since (x^2-4)^2>0 on domain, sign of f′ is sign of -x. So increasing on (-∞,-2) and (-2,0); decreasing on (0,2) and (2,∞). Thus x=0 is a local maximum (inc→dec). |
| Concavity | Differentiate f′=-6x(x^2-4)^{-2}: f″=-6(x^2-4)^{-2} + (-6x)(-2)(x^2-4)^{-3}(2x) ⇒ f″=-6/(x^2-4)^2 + 24x^2/(x^2-4)^3. Combine: f″ = [ -6(x^2-4) + 24x^2 ]/(x^2-4)^3 = [18x^2+24]/(x^2-4)^3 = 6(3x^2+4)/(x^2-4)^3. Numerator always positive, so sign depends on (x^2-4)^3: positive when |x|>2, negative when |x|<2. |
| Inflection points | Need sign change of f″ where f is continuous. f″ changes sign at ±2, but f is not defined there, so no inflection points. |
| Key function values (plausibility) | Near asymptotes: x→2^-, denominator negative small ⇒ 3/(x^2-4)→-∞ so f→-∞; x→2^+ ⇒ f→+∞. Similarly at -2: from left f→+∞, from right f→-∞. Sample values: f(3)=8/5=1.6, f(5)=24/21≈1.143 (approaching y=1). Inside: f(0)=0.25, f(1)=0, f(1.5)=(1.25)/( -1.75)≈-0.714. |
Choose a window (guided)
Because the graph blows up near x=±2, a window that shows both asymptotes and the horizontal asymptote is useful. Start with x from -6 to 6. For y, note values like f(3)=1.6 and f(0)=0.25, but also the vertical blow-ups; choose something like y from -6 to 6 so you can see the branches without flattening everything.
Clean sketch instructions (what to draw)
- Draw dashed vertical asymptotes at
x=-2andx=2, and dashed horizontal asymptotey=1. - Plot points:
(-1,0),(1,0),(0,1/4). - Use symmetry: sketch right side first, then mirror.
- On
(2,∞): decreasing and concave up, starting at+∞just right of 2, approachingy=1from above (confirmed byf(3)=1.6,f(5)≈1.143). - On
(0,2): decreasing and concave down, passing through(1,0), heading to-∞asx→2^-. - On
(-2,0): increasing and concave down, coming from-∞asx→-2^+up to the local maximum at(0,1/4). - On
(-∞,-2): increasing and concave up, approachingy=1from above asx→-∞, and going to+∞asx→-2^-.
Interpretation (write the behavior)
The function is even with vertical asymptotes at x=±2 and horizontal asymptote y=1. It crosses the x-axis at x=±1 and has a local maximum at (0,1/4). It increases on (-∞,-2) and (-2,0), decreases on (0,2) and (2,∞). It is concave down on (-2,2) and concave up on (-∞,-2) and (2,∞), with no inflection points because the only concavity sign changes occur at discontinuities.
Guided Example 2: Logarithm with a vertical asymptote and one critical point
Function: g(x) = x - ln(x)
Two-column analysis table
| Feature | Evidence / computation / conclusion |
|---|---|
| Domain | ln(x) requires x>0. Domain: (0,∞). |
| Intercepts | y-intercept: none (x=0 not in domain). x-intercept: solve x - ln(x)=0 (no simple closed form). Use reasoning: at x=1, g(1)=1; at x=0.1, g(0.1)=0.1-(-2.302)=2.402; values are positive, suggesting no x-intercept. |
| Symmetry | No even/odd symmetry (domain is only positive x). |
| Asymptotes | As x→0^+, ln(x)→-∞ so -ln(x)→+∞, hence g(x)→+∞. This indicates a vertical asymptote at x=0 (from the right). No horizontal asymptote since g(x)~x as x→∞. |
| Critical points | g′(x)=1-1/x=(x-1)/x. Set g′=0 ⇒ x=1. (Derivative undefined at 0 but 0 not in domain.) |
| Increasing/decreasing | For 0<x<1, x-1<0 so g′<0 ⇒ decreasing. For x>1, g′>0 ⇒ increasing. So x=1 is a local minimum. |
| Concavity | g″(x)=d/dx(1-1/x)=1/x^2>0 for all x>0 ⇒ concave up everywhere on domain. |
| Inflection points | None (concavity does not change). |
| Key function values (plausibility) | g(1)=1 (minimum). g(2)=2-0.693=1.307. g(5)=5-1.609=3.391. Near 0: g(0.2)=0.2-(-1.609)=1.809, g(0.05)=0.05-(-2.996)=3.046 (rising to +∞ as x→0^+). |
Choose a window (guided)
This graph has a right-hand vertical asymptote at x=0 and grows roughly like x for large x. A practical first window is x from 0 to 6. For y, since g(0.05)≈3.05 and g(5)≈3.39, a window like y from 0 to 5 shows the minimum at (1,1) and the rise near 0 without extreme blow-up. If you want to see more of the asymptote behavior, shrink x closer to 0 (e.g., 0<x<1) and increase the y-maximum.
Clean sketch instructions (what to draw)
- Draw the vertical asymptote line at
x=0(dashed), noting the graph exists only forx>0. - Plot the minimum point
(1,1). - On
(0,1), draw a decreasing, concave-up curve coming down from+∞toward(1,1). - On
(1,∞), draw an increasing, concave-up curve rising from(1,1)(check heights withg(2)≈1.307,g(5)≈3.391).
Interpretation (write the behavior)
The function is defined only for x>0 and has a vertical asymptote at x=0 with g(x)→+∞ as x→0^+. It decreases on (0,1), reaches a global (and local) minimum at (1,1), then increases on (1,∞). It is concave up everywhere, so the curve bends upward throughout and has no inflection points. The function stays positive on its domain (consistent with sampled values), so it does not cross the x-axis.
Verification habits: quick plausibility checks that catch common sketch errors
1) Check one value per interval
After you split the domain by asymptotes/critical points/inflection points, pick at least one x-value in each interval and compute f(x). This prevents mistakes like drawing the curve above the x-axis when it should be below, or placing a local extremum at the wrong height.
2) Check behavior near vertical asymptotes from both sides
For rational and log-based functions, the direction of blow-up matters. Use a one-sided test (e.g., plug in x=2.1 and x=1.9) to confirm whether the branch goes to +∞ or -∞.
3) Check end behavior with a large x-value
If you predict a horizontal asymptote y=L, compute f(10) or f(100) (as appropriate) to see whether the curve approaches from above or below. For oblique behavior, compare f(x) to the slant line at a large x.
4) Window selection is part of the mathematics
A “bad” window can hide key features (like a local extremum) or exaggerate others (like asymptote blow-up). Use your computed points and sample values to choose x- and y-ranges that show: intercepts, extrema, and at least one representative point on each branch.
