Rational Functions - 2/5/13

We learned about rational functions. To understand what this phrase meant, we remembered that a rational number was simply a fraction with integer values in both the numerator and denominator.

Example of a rational number : 3 / 5

Rational fractions are very similar to this ration above, except the integers are replaced with polynomials in the numerator and denominator.

Example of a rational function: y =  x  /  (x+1)

If you take

y =  x  /  (x+1)

and graph it on your calculator, it will look like this (with zoom standard):



The graph shows two curves that are separated by a gap at roughly (-1 , 0). That means it’s a discontinuous function.

But we don’t always need the graph to find most of the characteristics of the function, which include:  x-intercept, y-intercept, horizontal asymptote, vertical asymptote.

I’ll use y= x / (x+1) to show how to find each. (this section can be found in our book starting on page 254)

X-intercept: Plug in zero for y. Then look at the numerator and see how you could get zero by plugging in a number for x. In this case, because the term in the numerator is only x, we plug in 0. So the x-intercept is at (0 , 0)


Y-intercept: Plug in zero for all the x terms. It would look like this : 

y = 0 / (0 + 1)

And that becomes y = 0, so the y-intercept is also at (0 , 0). By looking at the graph on your calculator, you can estimate the intercepts fairly easily, but with other function equations it’s more difficult to do so.


Horizontal Asymptote: Examine the “end behavior” of the function as it curves along the x-axis towards infinity (∞) and negative infinity. What y value on the coordinate plane is it seemingly approaching, yet never touching? If you look at the graph of y= x / (x+1), the function seems to move towards y = 1 in the negative infinity direction, and also rises towards (yet never reaches) y = 1 in the positive infinity direction. Therefore, the horizontal asymptote is :

y = 1

Another way to check this mathematically is to plug in a large number for all x-values in the function. Use a number like 1,000,000:

y = 1,000,000 / (1,000,000 + 1)    =    0.999999

This shows that the function rises towards 1 in the x direction for infinity.

Vertical Asymptote: The vertical asymptote occurs when the denominator equals zero. Plug in a number for x in the denominator to get zero:

y= x / (x+1)    plug in -1 for the x in the denominator

y = x / (-1 + 1)   

y = x / 0

This means that the vertical asymptote exists at x =  - 1

Here is a picture from pg. 264 in our book explaining the occurrence of asymptotes:

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Homework Problems
Our homework on Tuesday was pg. 268 #1-6, and 7-10. We tried some of these problems on Monday.

9).  f(x) =   -2  /  (x - 5)

X-intercept: 

0 =   -2  /  (x-5)

*because there is no x variable in the numerator, this function has no x-intercept

Y-intercept:

Plug in zero for all x-values

y = -2 / (0 - 5)

y = -2 / -5

So the y-intercept is at (0 , 2/5)

Horizontal Asymptote:

The function appears to be heading towards positive/negative infinity in the x-direction at y = 0

Plug in 1,000,000 for all x-values

y = -2 / (1,000,000 - 5)

y = -0.000002

That is very close to y = 0

Vertical Asymptote:

Try plugging in a number for x in the denominator so that it will equal zero:

f(x)  = -2 / (5 - 5)

f(x) = -2 / 0

Because 5 works to make the denominator 0, the vertical asymptote is at x = 5

Alright, I hope this post helped you understand rational functions.

 -Jesse