Graphing. 1. Domain 2. Intercepts 3. Asymptotes 4. Symmetry 5. First Derivative 6. Second Derivative...

GraphingGraphing

1. Domain1. Domain

2. Intercepts2. Intercepts

3. Asymptotes3. Asymptotes

4. Symmetry4. Symmetry

5. First Derivative5. First Derivative

6. Second Derivative6. Second Derivative

7. Graph 7. Graph

DomainDomain

Denominator can not be zeroDenominator can not be zero

D=(-oo,-3)U(-D=(-oo,-3)U(-3,3)U(3,oo)3,3)U(3,oo)

Nonnegatives under even rootsNonnegatives under even roots

1-x1-x22 >= 0 >= 0

x not 0x not 0

D = [-1, 0) U (0, 1]D = [-1, 0) U (0, 1]

2

4

9y

x

21 xy

x

The domain of y =The domain of y =is x -1 or D = is x -1 or D =

A.A. TrueTrue

B.B. FalseFalse

3

2 6

( 1)

x

x

, 1 1,

DomainDomain


Nonnegatives under even rootsNonnegatives under even roots

1-x1-x22>=0 1-x>=0>=0 1-x>=0

and 1+x>=0 and 1+x>=0

D=[-1, 0) U (0, 1]D=[-1, 0) U (0, 1]

2

4

9y

x

21 xy

x

y = , the domain y = , the domain isisx >= 5. x >= 5. A.A. TrueTrue

B.B. FalseFalse

5x

InterceptsIntercepts

Set x = 0 and solve for ySet x = 0 and solve for y

Set y = 0 and solve for xSet y = 0 and solve for x

SymmetrySymmetry

f(-x) = f(x) => Even functionf(-x) = f(x) => Even function

Symmetry about the y axisSymmetry about the y axis

f(-x) = -f(x) => Odd functionf(-x) = -f(x) => Odd function

Symmetry about the originSymmetry about the origin

AsymptotesAsymptotes

Denominator = 0 when x = cDenominator = 0 when x = c

x = c is an asymptotex = c is an asymptote

y = c is an asymptotey = c is an asymptote

lim ( )x

f x c

First derivativeFirst derivative

Find the critical pointsFind the critical points

Max, min, or neitherMax, min, or neither

Increasing or decreasingIncreasing or decreasing

Second derivative Second derivative

ConcavityConcavity

Inflection pointsInflection points

GraphGraph

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Series2

1. Domain1. Domain

2. Intercepts2. Intercepts

3. Asymptotes3. Asymptotes

4. Symmetry4. Symmetry

5. First Derivative5. First Derivative

6. Second Derivative6. Second Derivative

7. Graph 7. Graph

2

2

( 1)

1

xy

x

DomainDomain


No square roots so Domain =No square roots so Domain =

Domain = R Domain = R

2

2

( 1)

1

xy

x

The domain is all real The domain is all real numbers or (-oo, +oo)numbers or (-oo, +oo)

A.A. TrueTrue

B.B. FalseFalse2

2

( 1)

1

xy

x



y intercepty intercept


x interceptx intercept

2

2

( 1)

1

xy

x

What is the y What is the y intercept?intercept?

2

2

( 1)

1

xy

x

What is the y What is the y intercept?intercept?

1.01.0

0.10.12

2

( 1)

1

xy

x



y intercepty intercept


x interceptx intercept

2

2

( 1)

1

xy

x

What is the x What is the x intercept?intercept?

2

2

( 1)

1

xy

x

What is the x What is the x intercept?intercept?

-1.0-1.0

0.10.12

2

( 1)

1

xy

x



y = 1y = 1


(x + 1)(x + 1)22 = 0 when x = -1 = 0 when x = -1

2

2

( 1)

1

xy

x

SymmetrySymmetry

f(-x) not equal f(x) => Not even f(-x) not equal f(x) => Not even functionfunction

f(-x) = (-x+1)f(-x) = (-x+1)22/(1+x/(1+x22))

-f(x) = (-x-f(x) = (-x22-2x-1)/(1+x-2x-1)/(1+x22) not equal f(-x)) not equal f(-x)

Not an odd functionNot an odd function

No symmetry about the originNo symmetry about the origin

2

2

( 1)

1

xy

x

AsymptotesAsymptotes

Where is the denominator zero?Where is the denominator zero?

lim ( )x

f x c

2

2

( 1)

1

xy

x

2

2

2 1lim

1x

x x

x

The denominator is The denominator is zero when x = -1.zero when x = -1.

A.A. TrueTrue

B.B. FalseFalse2

2

( 1)

1

xy

x

Horizontal asymptote at Horizontal asymptote at y = y =

2

2

2 1lim

1x

x x

x

Horizontal asymptote at Horizontal asymptote at y = y =

1.01.0

0.10.1

2

2

2 1lim

1x

x x

x

y =y =

y’ =y’ =

= =

2

2

( 1)

1

x

x

2 2

2 2

(1 )2( 1) ( 1) 2

(1 )

x x x x

x

2

2 2 2 2

( 1)[(1 )2 ( 1)2 ] ( 1)(2 2 )

(1 ) (1 )

x x x x x x

x x

What is the absolute What is the absolute value of both critical value of both critical points?points?

2

2

( 1)

1

xy

x

2

2 2 2 2

( 1)[(1 )2 ( 1)2 ] ( 1)(2 2 )

(1 ) (1 )

x x x x x x

x x

What is the absolute What is the absolute value of both critical value of both critical points?points?

1.01.0

0.10.12

2

( 1)

1

xy

x

2

2 2 2 2

( 1)[(1 )2 ( 1)2 ] ( 1)(2 2 )

(1 ) (1 )

x x x x x x

x x

Increasing?Increasing?

y’ =y’ =

y’(-2) < 0 y’(0) >0 y’(2) < 0 y’(-2) < 0 y’(0) >0 y’(2) < 0

2 2

( 1)(2 2 )

(1 )

x x

x

Where is it increasing?Where is it increasing?

A.A. (1, +oo)(1, +oo)

B.B. (-oo, -1)(-oo, -1)

C.C. (-1, 1)(-1, 1)

2

2

( 1)

1

xy

x

Y=Y=

y’ =y’ =

y’’ = y’’ =

2 2

( 1)(2 2 )

(1 )

x x

x

2

2

( 1)

1

xy

x

First derivativeFirst derivativeFind the critical pointsFind the critical points

x = -1, 1x = -1, 1y = 0, 2y = 0, 2Decreasing on (-oo, -1) U (1, +oo)Decreasing on (-oo, -1) U (1, +oo)Increasing on (-1, 1)Increasing on (-1, 1)Local min at x=-1 and local max at Local min at x=-1 and local max at x=1x=1

2

2 2

2(1 )'( )

(1 )

xf x

x

2

2

( 1)

1

xy

x

ConcavityConcavity

Find the inflection pointsFind the inflection points

x = 0 , -root(3), root(3)x = 0 , -root(3), root(3)

2

2 3

4 ( 3)''( )

(1 )

x xf x

x

2

2

( 1)

1

xy

x

ConcavityConcavity

Inflection pts at x = 0 , Inflection pts at x = 0 ,

y = 1, [root(3) + 1]y = 1, [root(3) + 1]22/4 , [-root(3) + /4 , [-root(3) + 1]1]22/4/4

1.87 0.131.87 0.13

2

2

( 1)

1

xy

x

3

Graphing. 1. Domain 2. Intercepts 3. Asymptotes 4. Symmetry 5. First Derivative 6. Second Derivative...

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Transcript of Graphing. 1. Domain 2. Intercepts 3. Asymptotes 4. Symmetry 5. First Derivative 6. Second Derivative...