Curve discussion (8 steps)
About this app
Full curve discussion in eight didactic steps — domain, symmetry, zeros, monotonicity, curvature, extrema, inflection points, asymptotes — cumulatively revealing for three functions (polynomial, rational with pole, rational with parabolic asymptote).
The procedure
Curve sketching examines a function systematically: (1) domain, (2) symmetry, (3) zeros, (4) monotonicity from the sign of f′, (5) concavity from the sign of f″, (6) local extrema where f′ changes sign, (7) inflection points where f″ changes sign, and (8) asymptotes. The order goes from coarse to fine. Together, the steps determine the shape of the graph.
Three example functions
The app analyses three functions. The polynomial x⁴ − x³ − 3x² + 5x − 2 = (x − 1)³(x + 2) has a minimum at x = −5/4 and a saddle point at x = 1. The rational function −(x − 2)²/(x + 2) has a pole at −2, a minimum at (−6, 16), a maximum at (2, 0) and the slant asymptote y = −x + 6. The third, 3x² + 3 − 6/x, has a pole at 0, an inflection point at (∛2, 3) and the parabola 3x² + 3 as an asymptote.
Things to try
Use “Next” and “Back” to go through the eight steps in order; each one adds marks to the graph and a calculation to the result panel. With “All on”, “All off” and clicks on single steps you can show just the parts you want, for example only extrema and inflection points. In the second function, notice that f″ does change sign, but only across the pole, so there is no inflection point.
Frequently asked questions
- What are the steps of curve sketching?
- Typically: domain, symmetry, zeros, monotonicity, concavity, extrema, inflection points and asymptotes or end behaviour. Some courses add the y-intercept or a table of values. The goal is a sketch of the graph that shows all key points.
- What is a saddle point of a function?
- A saddle point (or stationary inflection point) is an inflection point with a horizontal tangent. There f′ = 0 but f′ does not change sign, so it is not an extremum. An example is (x − 1)³(x + 2) at x = 1.
- When does a rational function have a slant asymptote?
- When the degree of the numerator is exactly one more than the degree of the denominator. Polynomial division then gives a linear function plus a remainder that tends to 0 as |x| → ∞. For −(x − 2)²/(x + 2) the asymptote is y = −x + 6.
- What does the app show for each function?
- For each of the three functions it shows all eight steps with calculation and result, such as derivatives, sign intervals and coordinates. Matching marks appear on the graph, and you can switch each step on or off individually.
Subject: Analysis (one-dimensional)
More from Analysis (one-dimensional)
- Derivative as a sequence of secants
- Bounded growth
- Units of an integral: rate times time
- Exponential growth and decay
- Optimisation: the largest rectangle
- Area between curves
- Area function is an antiderivative
- Sequential continuity
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