Curve Sketching Questions

[Panphobia]

Tomorrow I am having a curve sketching test in calculus and I was having some problems. Firstly the functions

code:

(x-1)/x^2 and 2x/(x^2+1)

the Horizontal Asymptotes should be y = 0 but there is an x - intercept on both, I do not know what happens to make the Asymptote permeable.

[Dreadnought]

We define an asymptote of a curve to be a line such that the distance between the curve and the line tends to zero as the curve tends to infinity. We don't require that the curve never cross the line (though we might ask that it does not cross it infinitely many times). The reason you might be accustomed to the idea that the curve never crosses the asymptote is that for curves given by continuous functions, the curve cannot cross a vertical asymptote. (Because a function f(x) takes on at most one value for any given x, so a continuous function could not cross the asymptote then come back to almost touch it). (not sure if that was convincing...)

Anyway, the same does not hold for horizontal asymptotes since you can have a continuous function f(x) such that f(x1) = f(x2) and x1 != x2 (basically 2 x's for one y but not 2 y's for one x is what I'm trying to say).

So as long as your curve tends to a line as you go to infinity that line is an asymptote (regardless of the behavior away from infinity).

[Panphobia]

hmmmm ok, now our teacher only taught us how to graph polynomial and rational functions using calculus. He said he will put other functions on the test such as e^x, ln, exponential, sinusoidal, or root. How would you graph those types using calculus, like i know calculating the x and y-intercepts would easy, but what else would you need to find with calculus in order to graph a function other than a rational and polynomial.

[Dreadnought]

Intercepts, extreme points (local max and min), general shape of the function, asymptotes and any other obvious point of interest (ex: tan(pi/4) = 1 ) are probably all you're expected to show.

Exponential and logarithmic functions have no extreme points, but you should know where the intercepts are and how they behave as they go to infinity (the shape is simple).
You probably already know quite a bit about trig functions, make sure the shape is right your intercepts are labeled and that your functions have the right local extreme points ans/or asymptotes (I like to make sure tan(pi/4) lines up with 1 on the y-axis).
Roots are polynomials mirrored on the line x = y (so y = f(x) = sqrt(x) <=> x = g(y) = y^2), mark the extreme points (the shape is usually pretty easy).

Teachers are usually pretty easygoing, it's called a sketch for a reason. Just think about the shape and any distinguishing feature of the function (you should have a general idea of these for exp(x), log(x), sin(x), tan(x), x^n and x^(1/n) ).

Note that I don't talk about cos nor inverse trig functions since they can be found using sin and tan.

Also, if you really don't know what to do, plug in values of x, draw the points and try to connect the dots on your graph (remember that almost all these functions are smooth and beware of things like sneaky extreme points between your dots).

[Panphobia]

hmmmmmm so back to polynomials, if I want to figure out whether the concavity of a function is up or down what do I sub into the second derivative? I know that if it is a rational, you just sub in points that are to the left and to the right of the VA, but for polynomial what do you sub? Is it just finding the end behaviour?

[Dreadnought]

Concavity is really just the slope of the first derivative of a function. Positive concavity is referred to as concave up and negative concavity is referred to as concave down, but it really just means that the slope is increasing or decreasing. How would you find out if the slope of the first derivative is positive or negative at some point?