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Comprehensive Guide to Curve Sketching: Analyzing Graphs Step-by-Step

Discover the essential steps to analyze and sketch curves effectively, covering intercepts, symmetry, domain, range, continuity, and asymptotes. Learn to identify maximums, minimums, and points of inflection, while understanding how to determine increasing and decreasing behaviors. Uncover the significance of the first and second derivative tests, and master the techniques to sketch a graph by hand without a calculator. This guide is perfect for students aiming to improve their graph analysis skills in calculus and mathematics.

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Comprehensive Guide to Curve Sketching: Analyzing Graphs Step-by-Step

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  1. Section 3-6 Curve Sketching

  2. Steps to Analyze a Graph: a) Intercepts and symmetry b) Domain and range (continuity) c) Asymptotes d) maximums and minimums e) Increasing & decreasing • Points of inflection and Concavity • graph

  3. Intercepts • Intercepts: x-intercept: when y = 0 y-intercept: when x =0

  4. Symmetry About the y-axis: • Replace every x with –x if the function is Symmetric about the y-axis (all exponents are even) About the origin: • Replace every x with –x if the function is symmetric about the origin (all exponents are odd) • About the x-axis: • not a function

  5. Asymptotes • Only occur in rational functions • Vertical: set denominator equal to zero • Horizontal: take the limit as x approaches infinity • Slant: occur when the degree in the numerator is one higher than the denominator • Use long division • Rewrite function as y = mx + b + remainder • Remainder tends to zero as x approaches infinity, the line y = mx + b is the asymptote

  6. Horizontal Asymptotes • BOBO BOTN EATS DC • Bigger on bottom: y = 0 • Bigger on top: none • Exponents are the same: divide coefficients

  7. Maximums and Minimums Use the first or second derivative test to find the x values Substitute x into the original equation to obtain points

  8. Increasing and Decreasing • Find critical points • 1st derivative test • Positive—increasing • Negative—decreasing

  9. Inflection Points Inflection points: Set 2nd Derivative equal to zero test for change in concavity

  10. Concavity 2nd derivative test Positive – concave up Negative- concave down

  11. Sketch the curve which has the following: relative max relative min increasing on and decreasing on concave up concave down point of inflection

  12. 2.) Sketch the graph of no calculator! • Intercepts and symmetry b) Domain and range (continuity)

  13. 2.) Sketch the graph of • Asymptotes • maximums and minimums

  14. 2.) Sketch the graph of e) Increasing & decreasing

  15. 2.) Sketch the graph of f) Points of inflection and Concavity

  16. 2.) Sketch the graph of g) Graph

  17. 3.) Sketch the graph of no calculator! • Intercepts and symmetry • Domain and range (continuity)

  18. 3.) Sketch the graph of • Asymptotes • maximums and minimums

  19. 3.) Sketch the graph of e) Increasing & decreasing

  20. 3.) Sketch the graph of f) Points of inflection and Concavity

  21. 3.) Sketch the graph of g) Graph

  22. Homework Page 215 # 7, 8, 9, 13, 23, 24, 27, and 29 all parts

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