Advanced Graphing

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Advanced Graphing

The following text is for those interested in more complex graphing techniques, and therefore assumes a higher level of understanding of mathematics. Click here to go back to the easier part of the lesson.

The point of graphing is to express the essential nature of the function in a visual form. Where graphing helps is that it gives you an ability to "see" functions in your head without even touching paper, if you practice enough. I can do that with most common functions. What we'll discuss here is determining that essential nature and translating it into paper.

Determine the domain of the function

While the simplest of functions, such as the linear functions and the quadratic functions, are defined for all values of x, the more complicated functions may be defined only with some subregion of the x axis.

Take, for example, function 1/x: it is not defined for x = 0, because for x = 0, 1/x would involve division by zero, which is illegal. Note: As you can see, as x approaches zero, the function tends to get bigger and bigger. In calculus, they would say that it approaches plus infinity as x approaches zero from the right.


Graph of y = 1/x

Or, consider function : it is not defined for x < 0, because square root of a negative value does not exist. (strictly speaking, the square root of negative values is a complex number, which you will study later.)


Graph of y =

So, for any equation, you first need to find all values of x where the equation has a value. These values are called a domain of the function. Some hints: if your function involves a division operation, exclude all values of x which result in division by zero. If your function involves a square root or logarithm, exclude all values of x which result in the argument to square root or to logarithm being negative. This covers most of the domain restrictions, although there can be many more.

Examples
Function Domain Graph
sqrt%28+x-1+%29 The domain is all points where x-1 >= 0, that is, x > 1.
sqrt%28+x%2B1+%29+%2B+1%2Fx The domain is all points where
  • the square root of x+1 is defined, that is, x+1 >= 0, that is, x >= -1
  • 1/x is defined, that is, x != 0.
This leaves us with the set of all x where x >= -1 and x != 0.

Again, do not attempt to plot the function where it is not defined. This is a major mistake.

Determine the special points of the function

The special points of any function include:

  • Intersections with the x axis. They are the "roots" of the equation f(x) = 0. To find them, set f(x) = 0 and solve it as you would a regular equation.

Graph of 2sin(x)
Intersection with the Y axis. This is simply the value of f(x) when x is equal to 0, that is, it is the point (0, f(0)).
Graph of 2cos(x)
Boundaries of the domain All functions' domains have boundaries. The boundary is a point that separates points belonging to the domain from points not belonging to it. A special case of such boundaries is plus or minus infinity. The +- infinity are not points in a regular sense, but it is worth considering how the function behaves when it approaches infinity.
Graph of sqrt%28x%2B1%29%2B1%2Fx
Maximums and minimums . Collectively called extremums, these points are the locally highest and lowest points of the function's graph. In the following graphs, the maximums and minimums are marked.
Graph of 2cos(x)
Note that finding points of minumum and maximum would require you to do some calculus. If you do not know calculus yet, you sometimes may succeed by just being creative, so do not give up. For example, you know that the maximum value for sin(x) is 1, so to find the maximums you need to find all points where sin(x) = 1. Or, for parabolas, you know that the horizontal coordinate of the point of extremum lies exactly between the roots, because parabolas are symmetrical. Etc etc.

Set up the coordinate system around the special points

The special points are what makes the function interesting. So you want to make sure that for your coordinates, you have chosen a big enough area to include at least the best of them. For most functions you would be able to include all special points (except, obviously, for infinity). For some functions called periodic, however, there is an infinite number of special points. A case in point is the 2sin(x) function above: it waves around the x axis endlessly. For such functions, what you want to do is include a few special points so that their nature would become evident.

This part of your job is mostly about not dong stupid things. One of these stupid things would be to set up your coordinates so that the graph of the function lies wholly outside your drawing.

Plot these special points using circles of X symbols

Shall I say more? :)

Determine slopes and curvatures

Analyze the function and find out where it is sloping upwards, where it is sloping downwards, where it is concave, and where it is convex.

Sloping upwards

Sloping downwards

Concave function

Convex function

Plot some more points around the special points

After all this work, all you need is to just plot a few more points, and the curve will emerge all by itself. Just pick a few points with some reasonable interval, use your calculator, and plot these points.

Draw the line

Draw the line through the points, little by little.