Block days Wed, Thurs Feb 5-6, Notes on Holes in the Graph and Discontinuities

Post date: Feb 6, 2014 9:35:08 PM

Notes:

Most of graphs you have drawn so far are continuouse. A continuous graph is one where you could draw the whole thing without lifting up your pencil. The graphs of rational functions can have break in them called holes.

If a function have a vertical asymptote or a hole, it is discontinuous.

Holes occurs when both the numerator and denominator equal zero. Since dividing by zero is undefined. The value when both the numerator and denominator equals zero is undefined. There will be a hole in the graph.

Let's look at an example:

In this exmaple, since there is a hole at x=-5, it is a discontinued function.

There can be as many holes as there are zeros in the denominator that are also in the numerator. Let's look at another exmaple:

Notice how in this case, the denominator will equal zero when x equals -6 and -7, but the numerator will not equal zero at those points. Instead of holes, there will be vertical asymptotes when x=-6,-7. This gives us a discontinuous function.

Examples:

1. f (x) = Is this function discontinued? why?

The graph of this function will look like this:

Because it has a vertical asymptote, it is discontinuous.

2.f (x) = Is this function discontinued? Why?

The numerator and denominator both share the same factor (x-2), when x=2, both the numerator and denominator will equal to zero. So we know there will be a hole in the graph. Therefore, it's discontinued.

Other Resources:

Graphing a Rational Function (With a Hole)

HW: Due tomorrow, Skills 29-31 & 33 Rational Functions Worksheet, Assignment 6 (18-25) section 9.8 & Study for Quiz Skills 29-31 & 33