Discover Features of Rational Graphs

Graphs of Rational Functions

Why do they look like that??

Today we are going to explore WHY Rational Graphs look like they do, and how different parts of the expressions shape the graph.
Start with the parent graph y = 1/x
Graph it in Desmos on the next slide.
What do you notice??

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y = 1/x
​
There are 2 asymptotes:

y = 0 (the x-axis)

x = 0 (the y-axis)

there are no x-intercepts or y-intercepts

Open Ended

Why are there no x or y intercepts? What about the equation of the graph prevents them from happening?

​Now graph y = 1/(x-2)

Graph it on the next slide and find its key features

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Open Ended

What changed and what stayed the same?

​What about the equation

y = 1/(x-2)

moved the vertical asymptote to x = 2?

​Notice the word
UNDEFINED
in the table where the
Vertical Asymptote (VA)
shows up??
It's undefined because there is nothing in that space on the graph.

​Can you guess what the graph of
y = 1/(x + 4)

will look like? Where will the asymptotes be?

​Now graph it in Desmos on the next slide and check it out!

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​So now the vertical asymptote is at
x = -4!

The horizontal asymptote is still at
y = 0

​Notice the word UNDEFINED
shows up in the table

This represents the Vertical Asymptote. You can see the
empty space in the graph going
up and down!

Multiple Choice

Without graphing, where is the vertical asymptote of

y = 1/(x -12)

• x = 12

• x = -12

• y = 12

• y = -12

Math Response

Without graphing, where is the vertical asymptote of

y = 1/(x + 65)?

Be sure to start your answer with x=

Type answer here

Correct Answer: x=-65

Deg^°

Rad

What do you think will happen by adding another factor in the denominator??

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​Now we have 2 vertical asymptotes

x= -3 and x = 2

We still have a horizontal asymptote of y = 0

​What do you think will happen if we add a factor to the numerator??
Let's try it out!
Graph y = (x + 3)/(x - 2)

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​The vertical asymptote is still x = 2 but now the horizontal asymptote is y = 1
AND
there is an x-intercept (Root) at
x = -3


How does a root of x = -3 relate to the equation
y = (x+3)/(x-2)??

Multiple Choice

What are the key features of

y = (x - 7)/(x + 5)

• vertical asymptote x=7 and root of x=5

• vertical asymptote x=5 and root of x=-7

• vertical asymptote x=-7 and root of x=5

• vertical asymptote x=-5 and root of x=7

Multiple Choice

What are the key features of

y = (x + 11)/(x + 6)?

VA stands for Vertical Asymptote

• root of x = -11 and VA of

  x = -6

• root of x = 11 and VA of

  x = 6

• root of x = -6 and VA of

  x = -11

• root of x = 6 and VA of

  x = 11

Multiple Choice

What are the key features of

y = (x + 3)(x - 1)/(x + 9)

• VA x = -9

  Roots x = -3 and x = 1

• VA x = -9

  Roots x = 3 and x = -1

• Root x = 9

  VA x = -3 and x = 1

• Root x = -9

  VA x = 3 and x = -1

​Here is the graph of

​What effect did the multiplier of
5 in the numerator
have on the graph?

What effect on the graph do you think will happen if we move the 5 to the denominator? Graph it and see..

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​Now the horizontal asymptote is y = 0.2

What is 0.2 as a fraction??


Where the Horizontal Asymptote graph depends upon the DEGREE of the numerator and the DEGREE of the denominator. We'll talk more of this soon....

​What happens when you REDUCE a rational expression?

It actually does something surprising! Let's go check it out.

​Where do you expect the VA and the roots to be?

​The VA is at x = -3
The root is at x = 4


But what happened to the other
VA and the other root??

The (x + 2) reduced from the numerator and denominator, but leaves an effect on the graph

Since you can not have a VA and an x-intercept in the same place,
we have created a hole in the graph!

Let's go zoom in a little closer on the graph and take a look at it's table!

​The word UNDEFINED shows up twice in the table....

​Although -2 is undefined in the table, it is NOT a
VA. The graph shows up
at x = -2.
This is called a HOLE
in the graph.
This comes from a
COMMON TERM in the numerator and denominator! (x +2) is a common term.

Multiple Choice

Where is the hole?

y=(x+5)(x+9)/(x+5)(x-7)

find the common factor

• x = -9

• x = -5

• x = 7

Multiple Choice

Where is the hole?

y = (x - 8)(x + 3)/(x + 8)(x + 3)

• x = 8

• x = -8

• x = 3

• x = -3

Math Response

Where is the hole?

y = (x² + 8x + 12)/(x² + 3x -18)

Factor both polynomials to find your common terms. Start your answer with

x=

Type answer here

Correct Answer: x=-6

Deg^°

Rad

Math Response

Where is the hole?

y = (x² - 81)/(x² + 11x + 18)

Type answer here

Correct Answer: x=-9

Deg^°

Rad

Poll

What do you think you can do on your own after this lesson?

• I can only find the VA

• I can only find the hole

• I can find the VA and the hole

• I am totally lost in this lesson