Graphing. 1. Domain 2. Intercepts 3. Asymptotes 4. Symmetry 5. First Derivative 6. Second Derivative 7. Graph.

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Presentation transcript:

Graphing

1. Domain 2. Intercepts 3. Asymptotes 4. Symmetry 5. First Derivative 6. Second Derivative 7. Graph

Domain Denominator can not be zero D=(-oo,-3)U(-3,3)U(3,oo) D=(-oo,-3)U(-3,3)U(3,oo) Nonnegatives under even roots 1-x 2 >= 0 1-x 2 >= 0 x not 0 x not 0 D = [-1, 0) U (0, 1] D = [-1, 0) U (0, 1]

The domain of y = is x -1 or D = A. True B. False

The domain of y = is x -1 or D = A. True B. False

Domain Denominator can not be zero Nonnegatives under even roots 1-x 2 >=0 1-x>=0 1-x 2 >=0 1-x>=0 and 1+x>=0 and 1+x>=0 D=[-1, 0) U (0, 1] D=[-1, 0) U (0, 1]

y =, the domain is x >= 5. A. True B. False

y =, the domain is x >= 5. A. True B. False

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

Symmetry f(-x) = f(x) => Even function Symmetry about the y axis f(-x) = -f(x) => Odd function Symmetry about the origin

Asymptotes Denominator = 0 when x = c x = c is an asymptote y = c is an asymptote

First derivative Find the critical points Max, min, or neither Increasing or decreasing

Second derivative Concavity Inflection points Graph

1. Domain 2. Intercepts 3. Asymptotes 4. Symmetry 5. First Derivative 6. Second Derivative 7. Graph

Domain Denominator can not be zero No square roots so Domain = Domain = R

The domain is all real numbers or (-oo, +oo) A. True B. False

The domain is all real numbers or (-oo, +oo) A. True B. False

Intercepts Set x = 0 and solve for y y intercept Set y = 0 and solve for x x intercept

What is the y intercept?

1.00.1

Intercepts Set x = 0 and solve for y y intercept Set y = 0 and solve for x x intercept

What is the x intercept?

0.1

Intercepts Set x = 0 and solve for y y = 1 Set y = 0 and solve for x (x + 1) 2 = 0 when x = -1

Symmetry f(-x) not equal f(x) => Not even function f(-x) = (-x+1) 2 /(1+x 2 ) -f(x) = (-x 2 -2x-1)/(1+x 2 ) not equal f(-x) -f(x) = (-x 2 -2x-1)/(1+x 2 ) not equal f(-x) Not an odd function No symmetry about the origin

Asymptotes Where is the denominator zero?

The denominator is zero when x = -1. A. True B. False

The denominator is zero when x = -1. A. True B. False

Horizontal asymptote at y =

1.00.1

y = y’ = y’ ==

What is the absolute value of both critical points?

1.00.1

Increasing? y’ = y’ = y’(-2) 0 y’(2) 0 y’(2) < 0

Where is it increasing? A. (1, +oo) B. (-oo, -1) C. (-1, 1)

Where is it increasing? A. (1, +oo) B. (-oo, -1) C. (-1, 1)

Y= y’ = y’ = y’’ = y’’ =

First derivative Find the critical points x = -1, 1 y = 0, 2 Decreasing on (-oo, -1) U (1, +oo) Increasing on (-1, 1) Local min at x=-1 and local max at x=1

Concavity Find the inflection points x = 0, -root(3), root(3)

Concavity Inflection pts at x = 0, y = 1, [root(3) + 1] 2 /4, [-root(3) + 1] 2 /