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Section 3-6

Section 3-6. Curve Sketching. 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. Intercepts. Intercepts: x-intercept: when y = 0

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Section 3-6

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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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