3.6a A Summary of Curve Sketching

Intercepts, symmetry, domain & range, continuity, asymptotes, differentiability, extrema, concavity,infinite

limits at infinity, WHEW!

When you are sketching a graph, either by hand or calculator• You normally can’t show the entire graph. • You will have to choose what you show. • Graph f(x) = x3 – 25x2 + 74 x – 20

OR

X : [-2,5] Y: [-10,40] X : [-10,30] Y: [-1200,200]

Calculus can help you know highlights of graph

Ex 1 p. 210 Rational Function

Analyze and sketch graph of 2

2

2( 9)( )

4

xf x

x

2 2

20first derivative: '( )

( 4)

xf x

x

2

2 3

20(3 4)2nd derivative: ''( )

( 4)

xf x

x

Vertical asymptotes:

Domain:

x-intercepts:

all reals except x≠-2, 2

1st derivative:

2nd derivative:

(-3, 0), (3, 0)

y-intercept: (0, 9/2)

Horizontal asymptote

Possible pts of inflection:

x = -2, x = 2

y = 2

Critical number(s): x = 0 (2 & -2 not critical, domain problems)

None

Symmetry: w/ respect to y-axis

Ex 1 continued . . . And yes, I expect this much calculus work to be shown!

f(x) f’(x) f”(x) Characteristics of Graph-∞ < x < -2 Neg Neg Decreasing, concave downx = -2 Undef Undef Undef Vertical asymptote-2 < x < 0 Neg Pos Decreasing, concave upx = 0 9/2 0 Pos Relative min0 < x < 2 Pos Pos Increasing, concave upx = 2 Undef Undef Undef Vertical asymptote2 < x < ∞ Pos Neg Increasing, concave down

Draw an excellent graph, label all asymptotes, important points like x- and y-intercepts, and extrema

Ex 2 p.211 Rational FunctionAnalyze and graph 2 2 4

( )2

x xf x

x

1st Derivative:

2nd Derivative:

Domain:

x-intercepts:

y-intercept:

Vertical asymptote(s):

Horizontal asymptote(s):

Critical number(s):

Possible points of inflection:

Symmetry:

f xx x

x' ( )

( )

( )

4

2 2

f xx

' ' ( )( )

8

2 3

All real numbers except x ≠ 2

none

(0, -2)

x = 2

None

x = 0, 4

None

None

Ex 2 continued

f(x) f’(x) f”(x) Characteristics of Graph-∞ < x < 0 Pos Neg Increasing, concave downx = 0 -2 0 Neg Relative maximum0 < x < 2 Neg Neg Decreasing, concave downx = 2 Undef Undef Undef Vertical asymptote2 < x < 4 Neg Pos Decreasing, concave upx = 4 6 0 Pos Relative Minimum4 < x < ∞ Pos Pos Increasing, concave up

This doesn’t have a horizontal asymptote, but since the degree in numerator is one more than that of denominator, it has a slant asymptote. Do division to see what the equation of the slanted line is.

Rewriting it after division,

f x xx

( ) 4

2

So the graph approaches the slant asymptote of y = x as x approaches +∞ or -∞

Ex 3 p.212 Radical FunctionAnalyze and graph

1st Derivative:

2nd Derivative:

Domain:

x-intercepts:

y-intercept:

Vertical asymptote(s):Horizontal asymptote(s):

Critical number(s):

Possible points of inflection:

Symmetry:

f xx

x( )

2 2

f xx

' ( )( )

2

22 32

f xx

x' ' ( )

( )

6

22 52

All real #’s

(0, 0)

Same (0, 0)None

y = 1 (right) and y = -1 (left)

None

At x = 0

With respect to origin

f(x) f’(x) f”(x) Characteristics of Graph-∞ < x < 0 Pos Pos Increasing, concave upx = 0 0 0 Point of inflection0 < x < ∞ Pos Neg Increasing, concave down

12

1

-1

-4 -2 2 4

horiz asymp. y = -1

horiz asymp. y = 1

(0, 0) point of inflection

f x = x

x2+2

3.6a p. 215/ 3-33 mult 3, 39