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3.6 A Summary of Curve Sketching

Vertical asymptotes (Section 1.5)

x­intercepts and y­intercepts (Section P.1)Symmetry (Section P.1)

Domain and Range (Section P.3)Continuity (Section 1.4)

Differentiability (Section 2.1)Relative extrema (Section 3.1)

Concavity (Section 3.4)Points of Inflection (Section 3.4)

Horizontal asymptotes (Section 3.5)Infinite limits at infinity (Section 3.5)

Tools/Concepts useful in Sketching a Graph[p202]

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1. Determine the domain and range of the function.2. Determine the intercepts, asymptotes, and symmetry of the graph.3. Locate the x­values for which f '(x) and f "(x) are either zero or do not exist. Use the results to determine relative extrema and points of inflection.

NOTE: In these guidelines, note the importance of algebra (as well as calculus) for solving the equations f (x) = 0 , f '(x) = 0 , and f "(x) = 0 .

Guidelines for Analyzing the Graph of a Function [p 202]

Polynomial Functions

Rational Functions f (x) = p(x)h(x)

if the degree of p(x) exceeds the degree of h(x) by one, oblique (slant) asymptote (after long division)

Radical Functions f (x) = xn

Trigonometric Functionsomit

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examples: Analyze and Sketch. [p208 #26]1. y = (x3 ­ 3x + 2)­1

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1.y y' y" Conclusion

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examples: Analyze and Sketch. [p208 #16]

2. f (x) = x3x2 ­ 4

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2.

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2.f (x) Conclusionf '(x) f "(x)

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examples: Analyze and Sketch. [p208 #20]

3. g (x) = x 9 ­ x

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3.

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3.

g (x) Conclusiong'(x) g"(x)

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examples: Analyze and Sketch.

4. f (x) = 2x ­ 5x 53 43

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4.

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4.

f (x) Conclusionf '(x) f "(x)

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p208 #1­6, 25, 31, 35 7, 9, 11, 17 19, 21, 23, 52­58

Assignment:polynomial

rational

radical concepts

matching ; reasoning