Hole vs VA

Holes vs Vertical Asymptotes

Shae Camardo

Warm Up

What are the restrictions to the function: f(x) = (x-2)(x-5)/(x-2)(x-3)

Answer

The restrictions are 2 and 3. This is because they are the real numbers that give a value of zero in the denominator and are essentially not part of the domain. How does this relate to our lesson? Well, the hole for the function is x= 2 and the VA is x= 3. We'll elaborate upon these topics in the next few slides.

Holes:

What are Holes? Before we compare, let's define some terms. A hole is when a value of x sets both the denominator and the numerator of a rational function equal to zero. This demonstrates there is a hole in the graph; that is, a single point at which the function has no value

Holes (continued)

The x value of the hole for this graph would be 5. To find the y substitute the value of x into the function’s simplified expression

f(x)

f(x)=

Vertical Asymptotes

What is a Vertical Asymptote?

A vertical asymptote is a vertical line that seems to coincide with the graph of a function but it actually never meet the curve. A vertical asymptote of a function plays an important role while graphing a function. In this example the vertical asymptote is x=3.

VA Continued

How to find VA's

EXAMPLE:

Step 1: Simplify the rational function. (factor the numerator and denominator of the rational function and cancel the common factors).

Find vertical asymptotes of f(x) = (x + 1) / (x2 - 1). Step One: f(x) = (x + 1) / [ (x + 1) (x - 1) ] = 1 / (x - 1) Now, set the denominator to zero. (x - 1) = 0 x = 1 So x = 1 is the VA of f(x).

Step 2: Set the denominator of the simplified rational function to zero and solve. Here is an example to find the vertical asymptotes of a rational function.

Comparison

How Do They Compare?

Therefore, if a factor cancels with a factor in the numerator, then there is a hole where that factor equals zero. If a factor does not cancel, then there is a vertical asymptote where that factor equals zero.

When a factor in the denominator does not cancel, it produces a vertical asymptotes

Holes occur when factors from the numerator and the denominator cancel.

MINI QUIZ

Check what you know

MULTIPLE CHOICE EXPLANATION

wHY ARE THOSE THE ANSWERS?

f(x)=(x+2/x-1) For the holes there is nothing that would cancel out (top and bottom). Therefore, there are no holes. Since there is still a factor in the denominator that did not cancel, you can set the denominator to zero to find VA's x-1= 0 giving you the answer x=1, which is your VA