How do you find intercepts, extrema, points of inflections, asymptotes and graph #y=x^5-5x#?
First step: factoring!
The y-intercept is (0, 0), since y(0) = 0.
To find the x-intercepts, factor the polynomial completely.
This gives us
The first equation is already solved. There is an x-intercept at (0, 0).
The second equation has two solutions.
Now you have the x-intercepts. Put them on a graph.
Next step: extrema. Take the first derivative.
Letting y = f(x), we have
Now f' is defined for all x and is zero only at
These are your critical values (possible extrema).
Find their y-coordinates, and add them to your graph.
Inflection points? Where is f'' equal to zero? Does f'' change sign at that location? If so, it's an inflection point.
A polynomial has no asymptotes.
You probably will not need to add more points in order to finish your graph, but perhaps plot the graph for x = -2 and x = 2.
These should complete it.