Rational Functions

Rational Functions

Rational Functions are a division among two polynomials
They have asymptotes:
vertical
horizontal
oblique*

Characteristics

Vertical Asymptotes

Horizontal Asymptotes

If the degree of P(x) and Q(x) are equal

If the degree of P(x) is less than Q(x)

​If the ​degree of P(x) is greater than Q(x)

Empty Points

Multiple Choice

What is the horizontal asymptote of the function given?

• y = 2

• x = 1

• x = 2

• y = -1

Multiple Choice

Which asymptote(s) are determined by looking just at the denominator?

• vertical

• horizontal

• end behavior

• none

Multiple Choice

What are the equation of the asymptotes?

• x=-3, y=1

• x=3, y =1

• x=-3, y =-1

• x=3 y=-1

Multiple Choice

What are equations of the asymptotes?

• x=-1 y = -3

• x=1 y = -3

• x=-1 y =3

• x=1, y=3

Multiple Choice

What is the Vertical Asymptote?

$$y=\frac{x-3}{x-6}$$  

• x= -5
• x= 5
• x= 6
• x= -6

Multiple Choice

Select the option with the right equations of the asymptotes

• x=-2 and y=-1
• x=-2 and y=1
• x=2 and y=1
• x=2 and y = -1

Multiple Choice

How many vertical asymptotes does the function have?

• None

• 1

• 2

• 3

Multiple Choice

Which of the following represents the vertical asymptote(s) for the function below?
$$f\left(x\right)=\frac{5}{x^2\left(x-4\right)\left(x+2\right)}$$  

• $$x=0,-4,2$$  

• $$x=4,-2$$  

• $$x=0,4,\ -2$$  

• $$x=-4,2$$  

Multiple Choice

Which of the following represents the graph of $$f\left(x\right)=\frac{1}{x^2-16}$$  ?

Math Response

What is the horizontal asymptote to this function?

Type answer here

Correct Answer: y=4

Deg^°

Rad

Math Response

What is the horizontal asymptote?

Type answer here

Correct Answer: y=1

Deg^°

Rad

Multiple Choice

Find the horizontal asymptote.

• None

• y=-2

• y=2

• y=0

Math Response

What is the horizontal asymptote of this function?

Type answer here

Correct Answer: y=\frac{1}{2}

Deg^°

Rad

Multiple Choice

The horizontal asymptote equals zero when:

• the degree of the numerator and denominator are equal

• the degree of the numerator is less than the degree of the denominator

• the degree of the numerator is greater than the degree of the denominator

• the numerator equals zero

Multiple Choice

Which asymptotes are determined by looking at the denominator of a function? 

• vertical

• horizontal

• slant

• none

Math Response

What is/are the vertical asymptote(s)?

Type answer here

Correct Answer: x=3

Deg^°

Rad

Multiple Choice

What's the vertical asymptote of the function?

• x = 5

• x = 0

• x =5 , x = -5

• x = 25

​Domain

​Range

​Domain and Range with hole(s)

Multiple Select

In finding the domain of the rational function

$$r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$$  where p(x) and q(x) are polynomial functions, all real numbers are allowed for x except . . .
(Check all that applies.)

• The ones that would make the numerator zero. $$p\left(x\right)=0$$  

• The ones that would make the denominator zero.  $$q\left(x\right)=0$$  

• The ones that would make the function zero. $$r\left(x\right)=0$$  

• The ones that would make the function undefined.

Multiple Choice

What is the domain of 

$$y=\frac{1}{x}$$  ?

• $$x\ne-1$$  

• $$x\ne0$$  

• $$x\ne1$$  

• $$x\ne y$$  

Multiple Choice

What is the range of 

$$y=\frac{1}{x}$$  ?

• $$y\ne-1$$  

• $$y\ne0$$  

• $$y\ne1$$  

• $$y\ne x$$  

Multiple Choice

Find the domain of 

$$y=\frac{2}{x+1}$$  .

• $$x\ne-1$$  

• $$x\ne0$$  

• $$x\ne1$$  

• $$x\ne2$$  

Multiple Choice

Find the range of 

$$y=\frac{2}{x+1}$$

• $$y\ne-1$$  

• $$y\ne0$$  

• $$y\ne1$$  

• $$y\ne2$$  

Multiple Choice

The domain of 

$$y=\frac{x-1}{x+2}$$  is

• $$x\ne-2$$  

• $$x\ne-1$$  

• $$x\ne1$$  

• $$x\ne2$$  

Multiple Choice

The range of 

$$y=\frac{x-1}{x+2}$$  is

• $$y\ne-2$$  

• $$y\ne-1$$  

• $$y\ne1$$  

• $$y\ne2$$  

Multiple Select

What is the domain of 

$$y=\frac{\left(x-1\right)\left(x+4\right)}{\left(x+2\right)\left(x-3\right)}$$ ?

(Check all that applies.) 

• $$x\ne-2$$  

• $$x\ne-1$$  

• $$x\ne3$$  

• $$x\ne4$$  

Multiple Choice

What is the domain of the rational graph?

• Domain:

  (-∞,1)U(1,∞)

• All real numbers.

• Domain:

  (-∞,3)U(3,∞)

Multiple Choice

What is the Domain of the graph?

• Domain: (-∞,-3)U(-3,∞)

• All real numbers

• Domain: (-∞,1)U(1,∞)

Multiple Choice

What is the Range of the graph?

• $$\left\{x\left|x\in\Re\ ,\ x\ne-3\ \right|\right\}$$

• All real numbers

• $$\left\{y\left|y\in\ \Re\ ,\ y\ne4\right|\right\}$$

Multiple Choice

Find the domain for the function $$f\left(x\right)=\frac{2x-1}{3x+6}$$

• x cannot equal to 2

• x cannot equal to -2

• x cannot equal to 3

• All real numbers

Multiple Choice

Find the range for the function $$f\left(x\right)=\frac{2x-1}{3x+6}$$

• x cannot equal to -2

• y cannot equal to 2/3

• y cannot equal to -2/3

• All real numbers

Multiple Choice

Find the domain of the function. $$f\left(x\right)=\frac{x}{2x+4}$$

• x cannot equal to 2

• x cannot equal to 4

• x cannot equal to -2

• All real numbers

Multiple Choice

Find the range of the function. $$f\left(x\right)=\frac{x}{2x+4}$$

• x cannot equal to 1/2

• y cannot equal to 1/2

• x cannot equal to -2

• All real numbers

Multiple Choice

Find the domain of the function $$f\left(x\right)=\frac{x^3}{x+8}$$

• x cannot equal to 2

• x cannot equal to -2

• x cannot equal to -8

• All real numbers

1) Graph all the asymptotes (vertical and horizontal)
2) Plot any empty point (hole) in the plane
3) Evaluate your function for two values at each side of each vertical asymptote

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​Graphing