Objectives:
- Students will explore oblique asymptotes using long division and a
graphical approach
- Students will review the procedure for graphing rational equations
- Students will solve and graph real world rational function problems
(reservoir acid problem and can cost problem)
PA Math Standards:
- 2.8.11 Algebra and Functions
- D. Formulate expressions, equations, ... to model routine and
non-routine problems
- Q. Represent functional relationships in tables, charts and graphs
- R. Create and interpret functional models
- S. Analyze properties and relationships of functions (e.g. rational)
- 2.11.11 Concepts of Calculus
- A. Determine maximum and minimum values of a function over a specified
interval
- B. Interpret maximum and minimum values in problem situations
- 2.3.11 Measurement and Estimation
- A. Select and use appropriate units and tools to measure to the degree
of accuracy required in particular measurement situations.
- C. Demonstrate the ability to produce measures with specified levels of
precision
Prerequisite Skills:
- Students have explored all forms of basic rational functions
- Students have graphed all forms of 1/x
- Students have also graphed all forms of the translated rational function
a(1/(x-h))+k
- Students understand how to:
- Find and plot x and y intercepts
- Determine symmetry by replacing x with -x
- Reduce rationals to lowest terms
- Find both vertical and horizontal asymptotes
- Determine points, if any, at which rational graphs intersect the
horizontal asymptote
- Locate and describe holes in rational graphs
- Students can perform polynomial long division
- Students have prior knowledge in solving mixture problems
Materials:
- TI-83 Graphing Calculators
- Graph Paper
- Diet Coke cans
- Tape Measure/Rulers (centimeters/inches)
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References:
- Prentice Hall Precalculus (Graphing and Data Analysis)
- HRW Advanced Algebra
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Oblique Asymptotes:
By this point, you should have the skill to find vertical and
horizontal asymptotes. For the case where the numerator is one degree larger than the
degree of the denominator, we can use long division to locate the oblique asymptote. We
will also explore the case where the numerator is greater than the denominator by more
than one degree.
Procedure
| Observe the rational function to the right: |
 |
| Since the rational function f(x) is improper, we can use long division
to find any horizontal oblique asymptote. Observe the long division process: |
 |
| As x approaches positive or negative infinity, the remainder will
approach zero. 6/(x-1) goes to 0 Why? |
Since 6/(x-1) goes to 0 , then f(x) approaches x+4. We can conclude
that the graph of the rational function has an oblique asymptote at the line y=x+4. We
will observe this graphically using the TI-83 calculator! |
|
| Graphing Calculator Graphs |
Explanation |
  |
These graphs represent f(x) near the vertical asymptote. The first graph is with the
points plotted in dot mode and the second is with the points in connected mode. |
  |
This graph shows our function as we move far right past the origin. Notice that for an
x value of 50, our function goes to approximately 54.12 which is slightly above the
oblique asymptote y=x+4 (for x=50 y=x+4 yields y=54). No matter how far we go towards the
right, our function will always be slightly above the line y=x+4 |
  |
This graph shows our function as we move far left prior to the origin. Notice that for
an x value of -50, our function goes to approximately -46.12 which is slightly below the
oblique asymptote y=x+4 (for x=-50 y=x+4 yields y=-46). No matter how far we go towards
the left, our function will always be slightly below the line y=x+4 |
|
Numerator Exponent is n
and the Denominator Exponent is > n+1 |
Finally, lets look at the case where the numerator is more than one unit
greater than the denominator. Observe the rational function to the right: |
 |
| Since the rational function g(x) is improper, we can use long division
like above to explore asymptote behavior. Observe the long division process: |
 |
| Again we can show that
the remainder approaches zero as x approaches positive or negative infinity. However,
unlike our previous example, this function approaches a non-linear function. Therefore,
there are no horizontal or oblique asymptotes! |
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Steps for Graphing Rational Functions
of the form
| Step 1: |
Locate the intercepts, if they exist, and plot them. In
R(x)=P(x)/Q(x), the x-intercepts satisfy the equation P(x) = 0. The y-intercepts occurs at
R(0) if there is one. |
| Step 2: |
Test for symmetry by replacing x by -x. If R(-x)=R(x), there is y-axis
symmetry because we have an even function. If R(-x) = -R(x), there is symmetry with
respect to the origin because we have an odd function. |
| Step 3: |
Reduce the function to lowest terms and find the real zeros of its
denominator. Each of these zeros is a vertical asymptote. For any real number a, when x-a
is a factor of both the numerator and the denominator of the function, the graph has a
hole at x=a. Place a circle over holes on the graph (if they exist) and refer to them with
a proper written description. |
| Step 4: |
Locate horizontal or oblique asymptotes, if they exist by using the
following list of checks:
- If function R(x) is such that n < m, then the graph will have a
horizontal asymptote y=0 (the x-axis)
- If function R(x) is such that n is greater than or equal to m, then
Polynomial long division can be used:
- If n=m, the quotient will provide a constant limit that will be an/bm.
- If n=m+1, the quotient obtained is of the form ax+b (linear function)
and the line y=ax+b is an oblique asymptote
- If n > m+1, the quotient obtained is a polynomial of degree 2 or
higher, and the function R(x) has neither a horizontal nor an oblique asymptote. The graph
will behave like the graph of the quotient however!
|
| Step 5: |
Create a table or chart and plot several points between the
x-intercepts and vertical asymptotes. Sometimes it is also helpful to plot a sign chart
showing whether the function is positive or negative between all of the x-intercepts and
vertical asymptotes. |
| Step 6: |
Graph the function R(x) |
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