Degree of the polynomial: odd Leading coefficient: negative Example Graph:
Practice makes perfect
Let's begin by sketching the graph of the polynomial function. Then we will describe the degree and the leading coefficient of f.
Sketching the Graph
We want to sketch the graph of a polynomial function with the given characteristics.
f is decreasing when x<- 1.5 and x>2.5
f is increasing when - 1.5
f(x)>0 when x<- 3 and 1
f(x)<0 when - 34
Let's begin by identifying the sections of the coordinate plane where the graph is above and below the x-axis. Since the function is positive when x<- 3 and when 14, here it will be below the x-axis.
Let's now identify the increasing and decreasing intervals. We are told that the graph decreases for values of x less than - 1.5 or greater than 2.5. We also know that the graph increases when x is greater than - 1.5 and less than 2.5.
Notice that now we can tell where the zeros, minimum, and maximum values of our function will be.
Zeros at - 3, 1, and 4 — at these points the function is changing sign of the values, this tells us the location of three of the real zeros.
Minimum at x=- 1.5 — at this point the function is changing from decreasing to increasing, this tells us that no other nearby points can have a lower y-coordinate.
Maximum at x=2.5 — at this point the function is changing from increasing to decreasing, this tells us that no other nearby points can have a greater y-coordinate.
We will sketch the graph of the function with only three real zeros. To do so, we will plot them on the coordinate plane, and then draw a curve that passes through those points. Keep in mind that there is a minimum at x=- 1.5, and a maximum at x=2.5.
Note that this is only one of the infinitely many graphs with the given characteristics. Any graph with three zeros at - 3, 1, and 4, which is decreasing and increasing in the given intervals, and above and below the x-axis in the corresponding sections will be correct.
Degree and Leading Coefficient
From the graph we can tell that f(x) approaches positive infinity as x approaches negative infinity. We can also see that f(x) approaches to negative infinity as x approaches to positive infinity.
f(x) → + ∞ as x → - ∞
f(x) → - ∞ as x → + ∞
From the up-down end behavior of the graph we know that our polynomial function has an odd degree and a negative leading coefficient.
