Rational Functions
- Author:
- Whit Ford
- Topic:
- Functions
Use his page to experiment with the parameters in the Rational Function
Each of the letters A, B, C, D, E, and F have a slider in the applet below which allows you to vary the value of that parameter. Experiment with each a bit to see how each affects the graph, then see if you can answer the questions below the applet.
Note that first equation in the applet matches the one above, and that the value of each slider appears in the second equation shown in the applet. You can use this equation with all of the values filled in to analyze the behavior of the graph.
The orange C slider controls the degree (the highest exponent of the variable once everything is multiplied out) of the numerator. Experiment with three possible situations: where the degree of the numerator is smaller, equal to, or larger than that of the denominator. How does the appearance of the graph change in each of these situations, and why?
The grey A slider controls the value of the constant term on the left of the numerator, and the grey D slider controls the constant term on the left of the denominator. What affect do these two sliders have on the appearance of the graph, and how does that affect change depending on the value of C? Why?
What effect do the B, E, and F sliders have on the graph? Why?
Try moving the green point so that it is no longer between the other two points along the x-axis.
What happens when:
- the denominator goes to zero (because one of its factors is zero)?
- the numerator goes to zero?
- the green point is on top of the red or the blue point on the x-axis? Why does this happen?
- A is zero? Why does this happen?
When do you see:
- a horizontal asymptote? What determines the y-value of the horizontal asymptote?
- a diagonal asymptote? How can you change its slope? What determines the slope of the asymptote? Why?
Can you create a graph that has:
- a horizontal asymptote below the x-axis?
- only one vertical asymptote?
- a central section that crosses the x-axis? Under what conditions can the central section cross the x-axis, and when will it not?
If you wish to use other applets similar to this, you may find an index of all my applets here: https://mathmaine.com/2010/04/27/geogebra/