To graph a function, you have to find some important points which belongs to the graph, to check if the function is increasing or decreasing, to verify if the function is continuous, to see if the function is convex or concave.
Let's see if the function is continuous:
lim f(x) = lim (x^3-4) = limx^3-lim4 = inf, if x->inf
lim f(x) = -inf, if x->-inf
The function is continuous.
Let's see if the function is increasing or decreasing.
For this rason, we'll calculate it's first derivative:
f'(x) = 3x^2
Because f'(x)>0, no matter if x>0 or x<0, that means that f(x) is an increasing function.
For x=0, the function has an inflection point.
To check the aspect of the function, we have to calculate the second derivative:
f"(x) = 6x
For x<0. f"(x)<0, so f(x) is concave
For x>0, f"(x)>0, so f(x) is convex.
Now, we have to give values for x and we'll find values for y, so that to draw the graph of f(x), which passes through these points.
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