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Lesson 5.4 Exploring the graphs of Rational Expressions
Activity 1: Using Desmos.com, graph the following function:
1.
What is different about the graph of function compared to the
polynomial functions that we studied in unit 4?
x
2.
•
•
•
•
•
•
•
Use Desmos, Complete the following data table at right:
Compare your answers with your group
What happens to the graph
as x approaches -3 from the left
as x approaches -3 from the right
What is the domain of the function?
STOP FOR CLASS DISCUSSION
Take notes as needed on last pg. of lesson
y
0
-1
-2.0
-2.5
-2.7
-2.9
---------------------------------------------------------------Activity 2:
Using Desmos.com, graph the following function:
1. Speculate: Why it makes sense that this graph is
a line? (hint, look closely at the degrees of the top/bottom)
2.
Complete the following data table at right
3.
On the same graph, also graph f(x) = x + 4
4.
Speculate: Why it makes sense that this looks like
the same line? (hint, factor the numerator and simplify)
5.
x
0
1
2
3
4
5
Delete the first function on Desmos and complete the data table
again for the second function on its own. Are all of the y values the
same?
6.
What is the domain of each function?
7.
Are these two functions really the same?
• STOP FOR CLASS DISCUSSION
• take notes as needed on last pg of lesson
y
Lesson 5.4 Exploring the graphs of Rational Expressions
Activity 3:
Using Desmos.com, graph the following function:
1.
Speculate as to where any vertical asymptotes might be
2.
Complete the following data table:
3.
Which values for x result in an undefined value for y?
x
4
2
0
-2
4.
How are those values represented on the graph (asymptote or
hole)?
5.
-4
Factor the denominator of the function. How might you explain the
why one discontinuity is an asymptote while the other is a "hole"
6.
Discuss/confirm results with group
• STOP FOR CLASS DISCUSSION
y
Lesson 5.4 Exploring the graphs of Rational Expressions
Activity 1 Notes
• This graph is different because it is not continuous
• As x approaches -3 from the negative direction, the graph goes
to positive infinity
• As x approaches -3 from the positive direction, the graph goes
to negative infinity
• The domain of the function is all real numbers except x ≠ -3
• On the graph, the (invisible) line x = -3 is a Vertical
Asymptote
> An Asymptote represents the end behavior of a curved line
as it approaches either +/- infinity. The graph approaches
the asymptote but NEVER touches/crosses it.
Activity 2 Notes
• It makes sense that the graph is linear because the function
represents a 2nd degree divided by a 1st degree. We know from
unit 3 that the result must be a 1st degree (which is linear)
• It makes sense that f(x) = x + 4 is the same line because it is
equivalent to a simplified version of the original function
• The y values for x = 4 is NOT the same. For the original function
when x = 4, y is undefined, for the second function, y = 8
• The Two functions are NOT the same even though one is a
simplified version of the other. The original function has an Point
of discontinuity (Hole) at x = 4
> A point of discontinuity is a "gap" or "hole" in a graph that
would otherwise be continuous. It represents a domain
restriction where the y value is undefined.
• The domain of the original function is all real numbers except x ≠ 4
• The domain of the second function is all real numbers
Activity 3 Notes
• Vertical Asymptotes may exist at x = 4 and x = -4. These are
the "zeros" of the denominator of the original function
• x = 4 and x = -4 have undefined y values
• x = 4 is represented as a "hole" and x = -4 is represented as a
vertical asymptote
> A "hole" is the result when the factor that causes the
denominator to equal 0 also exists in the numerator (and
thereby can be cancelled out)
> An Asymptote is the result when the factor that causes the
denominator to equal 0 does not exist in the numerator