CURVE SKETCHING

section 3-A continued

Analyzing the graph of a function a) Domain and Range: All real numbers except ___b) Extrema and the intervals where increasing and

decreasing (first derivative test)c) Intercepts: where the graph crosses the x-axis

and the y-axis d) Inflection points and the intervals where concave

up and concave down (second derivative test)e) Symmetry

1. About the y-axis if even function2. About the origin if odd function

)()( xfxf )()( xfxf

Find all critical values f) Asymptotes- rational functions• Vertical: set the denominator equal to zero and verify

the limit tends to infinity

• Horizontal: Take the limit of the function as x approaches ±∞

• Slant: occur when the degree of the numerator is one higher than the degree of the denominator. Use long division or synthetic division to find the line

g) Graph- put it all together

4) Analyze the graph (parts a through g) then sketch 23 3)( xxxf

a) Domain and Rangeb) Extrema and the intervals where increasing

and decreasing (first derivative test)c) Intercepts: where the graph crosses the x-

axis and the y-axis d) Asymptotes- vertical, horizontal and slante) Inflection points and the intervals where

concave up and concave down (second derivative test)

f) Symmetry g) Graph- put it all together

5) Find the intervals where concave up and concave down if

12

24)(

2 x

xf32

2

)12(

576144)("

x

xxfand

6) Analyze the graph (parts a through g) then sketch

2

32)(

2

x

xxxf

HOME WORKPage 195 # 21, 24, 25, 36, 41, 48, and 49 worksheet 3-A