Rational functions?

Question

Can someone please tell me how to graph a rational function in terms I can understand? I sort of understand how to graph the x and y asymtotes, but beyond that im kind of lost... You can use the problem f(x)=x^2/x+5 as your example.

Wow, lots of big words O.o I think that helped though. I have a graphing calculator, but we have to sketch these by hand on our our test. And since I'm in an accelerated block class, we only get 70 minutes per lesson T-T.

Answer 1

I'm not sure if you are doing this by hand or by calculator. If its by hand, the first thing to do would be to find the limits, or the most obviously important parts of the graph. Since the denomenator is x - 5 and you cannot divide by 0, you know that when x=-5 the point is impossible to plot-- it either approaches infinity or negitive infinity. A quick test of plugging in x = -3 tells you that its approaching infinity on that side (and will eventually hit the point 0,0) so you make a curve where it approaches infinity.

The end would look something like this(although curved). Plug in a couple x values and it should be no problem



........|........|......... /
........\........|......../
..........\......|......./
......___\ _ |__/_____
.......-5.....0......5
.................|
++._......... |
++/..\.........|
+ /....|........|

Answer 2

Boy, you sure do ask a lot of a person. When I teach this topic fully it takes me 3 lessons of 75 minutes each!

But to sum it up, It is good you can state the asymptotes!

You also need to find the critical values by taking the first derivative of the function. Once you have the first derivative, set it to zero, and solve for x. This is usually not as hard as it looks.

Then, take the second derivative of the function.
Once you have that, there are a few things you need to do. First, plug the critical values that you found from the first derivative into the second derivative. If the result is positive, then the critical value is a minimum. If the result is negative, then the critical value is a maximum. If the result is zero, then the critical value is really a horizontal point of inflection.
Now, set the second derivative to zero and solve for other possible points of inflection. You will need to check the points in the viscinity of the PPOI's to determine if they are really POI's or if they are false POI's, and to determine the concavity of the graph.

Once you have all this data, you need to compile it into a graph. The asymptotes divide the cartesian plane into sections for you to graph, so draw those first.
Then, plot any x and y intercepts.
Then, plot any critical values and points of inflection.
I use the concavity from the second derivative to get an idea of the shape of the graph and sketch it.

I know this is probably not what you were hoping for in terms of an explanation, but I don't have 225 minutes to explain it to you. Sorry.

Anyway, I suggest you invest in a good graphing calculator.

Answer 3

What I do is first draw the asymptote, which is x = -5. you know your graph is going to be a hyperbola. I then just create a table of values using x values on each side of -5. Plot the points from the table and sketch the hyberbola.

there are probably specific instructions in your text for computing the center, the local minima and maxima, and foci of the curves. These would be helpful. Unfortunately, I would have to look those up myself, and I don't have a text handy

Answer 4

Another thing to do is see what happens as x approaches positive and negative infinity. In this case, the constant 5 becomes unimportant and we see that the line y=x is an asymptote. Also, see what happens as x gets very close to the asymptote x= -5 coming from both the left and the right.