How to graph cubic functions, quartic functions, and so on...?

Question

I don't understand how to graph functions like these and how can I tell how many real zeros there are?

Example: Graph the function: x^3+5

Please explain thoroughly because I have a huge amount of trouble understanding problems like these.

Answer 1

To give yourself an idea of a sketch, take several pairs of values... like (0,0) , (0,1), (1, 0), (1,2)... whatever. You'll see a pattern emerging.

However, with basic functions that very clearly involve something like x^2 or x^3, where you just add or take away something, you can just draw the underlying thing (x^2 is a parabola, x^3 is a curvy "S" going through the origin) and shift it by that amount. In the case of a parabola, something like (x-h)^2 +k is a parabola with "vertex" (the peak/trough point) at (h,k).

All else fails, there are graphing applets online that'll show you a few examples, which will give you a better idea. Here's one example: http://www.ies.co.jp/math/java/calc/sg_para/sg_para.html


(NOTE: (x,y) here means that x is the x-value corresponding to y, the y-value.)

Answer 2

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RE:
How to graph cubic functions, quartic functions, and so on...?
I don't understand how to graph functions like these and how can I tell how many real zeros there are?

Example: Graph the function: x^3+5

Please explain thoroughly because I have a huge amount of trouble understanding problems like these.

Answer 3

If you know calculus, you can find the extreme points, the points of inflection, and the zeroes, which is enough to make a very accurate graph. If you don't know calculus, you can find the zeroes. That doesn't tell you much. You can plug in numbers for x to get values for y, which will appear on your graph as (x, f(x)). Do a lot of these, and you might be able to make a fairly accurate graph. Obviously, you'll have to guess what the extrema and points of inflection are, but if you have enough points, that should be ok.

Answer 4

Any simple function with exponents above 3, either behaves the same way as either a quadratic or a cubic. If the greatest exponent is even, it behaves like a quadratic, except it grows much faster. If the greatest exponent is odd, it behaves like a cubic, except it grows much faster.

ex: x^4 + 1 behaves like x^2 +1, except it grows faster
x^7 + 1 behaevs like x^3 +1, except it grows faster

In terms of real zeros, if you end up taking the square root of a negative number, you know that there are going to be imaginary solutions, which cuts down (or eliminates) the possibility of real solutions.