SECTION 3-6Curve Sketching

Steps to Analyze a Graph:

a) Intercepts and symmetry

b) Asymptotes

c) maximums and minimums

d) Increasing & decreasing

e) Points of inflection

f) Concavity

g) graph

Intercepts• Intercepts:

x-intercept: when y = 0

y-intercept: when x =0

SymmetryAbout 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

)()( xfxf

)()( xfxf

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

Horizontal Asymptotes• BOBO BOTN EATS DC

• Bigger on bottom: y = 0• Bigger on top: none• Exponents are the same: divide coefficients

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)

Increasing and Decreasing

• Find critical points• First derivative test• Positive—increasing• Negative—decreasing increasing

increasing

Inflection Points

Inflection points:

Set 2nd Derivative equal to zero and solve

Test for changes in concavity

Concavity

2nd derivative test

Positive – concave up

Negative- concave down

1) Sketch the curve which has the following:relative max

relative min

increasing on and

decreasing on

concave up

concave down

point of inflection

( ,0) ),2(

(0,2)

(1,)

( ,1)

(0,4)

(2,0)

(1,1)

xxxf 63)( 2 2.) Sketch the graph of no calculator!

b) Asymptotes

a) Intercepts and symmetry

xxxf 63)( 2 2.) Sketch the graph of

c) maximums and minimums

xxxf 63)( 2 2.) Sketch the graph of

d) Increasing & decreasing

xxxf 63)( 2 2.) Sketch the graph of

e) Points of inflection

f) Concavity

xxxf 63)( 2 2.) Sketch the graph of

g) Graph

25

2)(

2 x

xxf3.) Sketch the graph of

no calculator!

b) Asymptotes

a) Intercepts

3.) Sketch the graph of 25

2)(

2 x

xxf

c) maximums and minimums

3.) Sketch the graph of

d) Increasing & decreasing

25

2)(

2 x

xxf

3.) Sketch the graph of 25

2)(

2 x

xxf

e) Inflection Points

3.) Sketch the graph of 25

2)(

2 x

xxf

f) Concavity

3.) Sketch the graph of

g) Graph

25

2)(

2 x

xxf