1. Section 3-6
Curve Sketching

2. Steps to Analyze a Graph:
Intercepts and symmetry
Asymptotes
maximums and minimums
Increasing & decreasing
Points of inflection
Concavity
graph

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

4. Symmetry About the y-axis: About the origin: About the x-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 derivative test to find the x values Substitute x into the original equation to obtain y-coordinate Points: ordered pair (x,y)

8. Increasing and Decreasing
Find critical points
First derivative test
Positive—increasing
Negative—decreasing
increasing
increasing

9. Inflection Points Set 2nd Derivative equal to zero and solve
Test for changes in concavity

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

11. 1) 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! a) Intercepts and symmetry
b) Asymptotes

13. 2.) Sketch the graph of
c) maximums and minimums

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

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

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

17. 3.) Sketch the graph of
no calculator!
a) Intercepts
b) Asymptotes

18. 3.) Sketch the graph of
c) maximums and minimums

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

20. 3.) Sketch the graph of
e) Inflection Points

21. 3.) Sketch the graph of
f) Concavity

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