The value f(c) is a relative minimum, or local minimum, of a function if f(c) is the least output of f around x=c. Likewise, the value f(d) is a relative maximum, or local maximum, of a function if f(d) is the greatest output of f around x=d.
Relative minimums and maximums of polynomial functions are also called turning points. This is because a function's graph turns from increasing to decreasing, or vice versa, at these points.
The graph of a polynomial function of degree n can have at most n−1 turning points. The function shown is a 3rd degree polynomial and has 2 turning points.Furthermore, a polynomial function with n real zeros must have at least n−1 turning points.
If a function has a symmetry, it is either even or odd. The symmetry is even when the graph is symmetric with respect to the y-axis, and odd when it's symmetric about the origin.
We can determine if a function has even or odd symmetry, using its function rule and the following relationships.
Simplify power and product
The figure below shows the graph of a polynomial function.
Use the graph to determine
Draw a graph of a polynomial function, f(x), having these characteristics.
Two of these points, (-3,0) and (3,0), lie in intervals where f(x) is decreasing. The third point, (1,0), is instead in an interval where the function is increasing. Using this, we can draw short sections of the function through the zeros.
When a function changes from increasing to decreasing, or vice versa, it does so at a turning point. From the given information, these changes occur at x=-1 and x=2. We can extend the already-drawn sections to these x-values.
Note that, because we don't have the exact coordinates of the turning point, their position is approximate. The function is decreasing both when x<-1 and when x>2. Since we know that, we can now draw the rest of the graph.