Question

How do i describe the shape of the graph of each cubic function by determining the end behavior and number of turning points?
for ex. y=3x^3-x-3

Answer 1

If the leading coefficient (coef. of x^3) is positive, the graph goes from on the left -inf to +inf. If it is negative, it goes from +inf (on the left) to -inf.
Check f'(x)=0. If there are two real and distinct solutions, there will be a maximum and a minimum. If the two solutions are coincident, then there is a turning point (neither max nor min). If there is no solution, the the graph goes monotonic inc. or dec.

You can also describe the zeroes by solving for them.

Answer 2

dy/dx and dy²/d²x

Answer 3

dy/dx = 9x²-1

Answer 4

when dy/dx=0, u will find its turning point
9x²-1=0
x²=1/9
x=(+ or -) 1/3
tat mean the truing point of the graph has x of -1/3 and -1/3.
to find the y of the turning point, simply input the value of x into the original function
in ur case:
when x=-1/3
y=3(-1/3)^3 +1/3 - 3
when x =1/3
y=3(1/3)^3 -1/3 - 3