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Calculus practice

Graphing from Function Properties worksheets

Analyze exact intercepts, critical points, inflection points, behavior intervals, concavity, and relative extrema before sketching a function.

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Find the selected properties, then sketch the function on the displayed grid.

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Read the whole displayed interval

Aligned axes, explicit windows, and separate rational branches make slope and shape claims visible rather than implied.

Learning objective

What this activity builds

Connect derivative sign information and exact feature coordinates to a complete graph on a stated window.

Worked example

Sketch f(x) = x^2 - 4 from its properties.

Intercepts (-2,0), (2,0), and (0,-4); minimum (0,-4); decreasing then increasing; concave up.

The roots and y-intercept locate the curve, while f'(x)=2x and f''(x)=2 determine its turn and concavity.

Teaching notes

Make the practice count

  • Treat poles as domain boundaries, not critical or inflection points.
  • Use derivative signs on every continuous interval.
  • Plot exact coordinates before shaping the curve.

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Choose a nonempty set of function families. Every graph uses a stated window and keeps discontinuous branches separate.

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Function families
Properties to find
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Math SheetNameScoreTeacherDateGraphing from Function PropertiesUse the listed properties to sketch each function on the displayed window.1. f(x) = -(x + 2)^2 + 4All real x.-6-5-4-3-2-1123456-10-8-6-4-2246810xyfFind: intercepts, critical points, inflection points, increasing decreasing,concavity, relative extrema.Use the blank grid to sketch f on the displayed window.2. f(x) = (x + 1)^2(x - 1)All real x.-6-5-4-3-2-1123456-10-8-6-4-2246810xyfFind: intercepts, critical points, inflection points, increasing decreasing,concavity, relative extrema.Use the blank grid to sketch f on the displayed window.Behavior intervals are global; the sketch uses the finite displayed window.Page 1 of 2math-sheet.com
Math SheetNameScoreTeacherDateGraphing from Function PropertiesUse the listed properties to sketch each function on the displayed window.3. f(x) = -(x - 1) - 4/(x - 1)All real x except x = 1-6-5-4-3-2-1123456-10-8-6-4-2246810xyfFind: intercepts, critical points, inflection points, increasing decreasing,concavity, relative extrema.Use the blank grid to sketch f on the displayed window.Vertical asymptote: x = 1. Slant asymptote: y = -(x - 1). Domainexcludes x = 1.4. f(x) = 2x - 2/xAll real x except x = 0-6-5-4-3-2-1123456-10-8-6-4-2246810xyfFind: intercepts, critical points, inflection points, increasing decreasing,concavity, relative extrema.Use the blank grid to sketch f on the displayed window.Vertical asymptote: x = 0. Slant asymptote: y = 2x. Domain excludes x= 0.Behavior intervals are global; the sketch uses the finite displayed window.Page 2 of 2math-sheet.com

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