help me with graphing part

[crisp skiff]

Try expanding the numerator, then use algebraic long division to see where that'll get you

[rough flax]

yeah or an alternative is using method of intervals

[hybrid tinsel]

hard

[idle kelp]

First of all, before i say anything about this, you can never graph such functions with 100% accuracy, whatever you do the graph will look a little different from the real graph

but if u take these steps, you will get a graph which is very close to the correct one:

1.horizontal asymptotes or oblique asymptotes (limits as x approaching infinty)

2.vertical asymptotes (limits when x approaching a certain number makes the y-value approach infinity)

3.derivatives (not necessarily but could help when graphing functions)

generally knowing these stuff can help you graph it

Lets solve an example together:

The very first thing we do is to see in which x-value the function has a vertical asymptote, in order to know this, you have to see where the denominator becomes zero (as you might have thought, for example 5/x when x->0+, 5/0+ is huge number, basically we consider these type of situations inf)

Now we know that the function should look something like this around x=1

Now its time to see how the function behaves when x is approaching infinity, in order to do this, we simplify the function:

And we get this simple looking function which is easier to analyze/work with, when x approaches infinity, based on limit rules the fraction approaches the number 3 (because 3x/x = 3), therefore when x is approaching infinity, the function basically shoud look like 2x+3

And when we draw the oblique asymptote(the new line), the function should get closer and closer to that when x is approaching infinity(the new part of the graph is the blue part!)

[idle kelp]

And this is our results of graphing it this way, you can make this more accurate by considering other things, like when the f’(x)=0, or when the function touches the x-axis and passes through it

And this is the correct graph! As you can see they are similar, but not that accurate, as i said, in order to make this more accurate, you gotta see where the derivatives are zero, where the functions roots are, and stuff like that

Tell me what steps you didn’t understand or might need more explanation to be clear, or anything that might be new for you

This was a very short explanation and i didnt really spend time on it cuz i have to make sure you generally understand it

So then we can get into the details…

[idle kelp]

☠️

im a student