1. 1 Sec 4.3 Curve Sketching

2. 2 Curve Sketching Problems Given: A function y = f(x). Objective: To sketch its graph.

3. 3 Steps (1)Find a “Frame” for the graph  Domain  Asymptotes – Horizontal, Vertical, Slant (2)Find out how the graph “wiggles”  Derivative – intervals of increase/decrease; max/min  Second derivative – intervals for concave up/down; point(s) of inflection (3)Sketch

4. 4 Example (1) Sketch Frame: Domain: Asymptotes: Starts hereEnds here Next Question: How does the graph wiggle between the two ends ?

5. 5 Wiggle: Derivative: 2 nd derivative: Final Step: Put the wiggly graph onto the Frame. + ++ –– ––

6. 6 Starts here Decreasing; Concave down Decreasing; Concave up Increasing; Concave up Increasing; Concave down Decreasing; Concave down Decreasing; Concave up Ends here A “twist” : Concavity changes – a point of inflection Graph rebounds after a dip – a local min A “twist” : Concavity changes – a point of inflection Local max A “twist” : Concavity changes – a point of inflection

7. 7 Example (2) Sketch Frame: Domain: Asymptotes: Starts hereEnds here Next Question: How does the graph wiggle within each of the three sections ? ? ? ? ? ? ? ? ? ? ? ?

8. 8 Wiggle: Derivative: 2 nd derivative:

9. 9 Example (3) Sketch Frame: Domain: Asymptotes: Starts hereEnds here Next Question: How does the graph wiggle within each of the three sections ? ? ? ? ? ? ? ? ? ? ? ?

10. 10 Wiggle: Derivative: 2 nd derivative:

11. 11 Example (4) Sketch Frame: Domain: Asymptotes: Starts here Ends here Next Question: How does the graph wiggle between the two ends ? ? ? ?

12. 12 Wiggle: Derivative: 2 nd derivative:

13. 13 Example (5) Sketch Frame: Domain: Asymptotes: Starts here Ends here Next Question: How does the graph wiggle within the two regions ? ? ? ? ? ? ?

14. 14 Wiggle: Derivative: 2 nd derivative:

15. 15 Example (6) Sketch Frame: Domain: Asymptotes: Repeat here Next Question: How does the graph wiggle in one of the regions ? ? ? ? Periodicity: ? Repeat here

16. 16 Wiggle: Derivative: 2 nd derivative: