A Summary of Curve Sketching

Lesson 4.6

How It Was Done BC(Before Calculators)

• How can knowledge of a function and it's derivative help graph the function?

• How much can you tell about the graph of a function without using your calculator's graphing?

Regis might be calling for this information!

Regis might be calling for this information!

Algorithm for Curve Sketching

• Determine domain, range of the function• Determine critical points

Places where f ‘(x) = 0

• Plot these points on f(x)• Use second derivative f’’(x) = 0

Determine concavity, inflection points

• Use x = 0 (y intercept) • Find f(x) = 0 (x intercepts)• Sketch

Recall … Rational Functions

• Leading terms dominate m = n => limit = an/bm

m > n => limit = 0 m < n => asymptote linear diagonal

or higher power polynomial

...

...

nnm

m

a x

b x

Finding Other Asymptotes

• Use PropFrac to get

• If power of numerator is larger by two result of PropFrac is quadratic asymptote is a parabola

( )( )

ry m x b

d x

Example

• Consider

• Propfrac gives

5 4

3 2

2 7

5 3 3

x x x

x x x

Example

• Note the parabolic asymptote

Other Kinds of Functions

• Logistic functions

• Radical functions

• Trig functions

/ 2

10( )

2 3 xh x

e

216y x x

1( ) cos cos 2

2f x x x

Assignment

• Lesson 4.6

• Page 255

• Exercises 1 – 61 EOO