Degree of the polynomial: even Leading coefficient: positive 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<0.5
f is increasing when x>0.5
f(x)>0 when x<- 2 and x>3
f(x)<0 when - 2
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<- 2 and when x>3, the graph here will be above the x-axis. Conversely, since the function is negative when - 2
Let's now identify the increasing and decreasing intervals. We are told that the graph decreases for values of x less than 0.5. We also know that the graph increases when x is greater than 0.5.
Notice that now we can tell where the zeros, minimum, and maximum values of our function will be.
Zeros at - 2 and 3 — at these points the function is changing sign of the values, this tells us the location of two of the real zeros.
Minimum at x=0.5 — 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.
We will sketch the graph of the function with only two real zeros. We will plot them on the graph and then draw a curve that passes through those points. Also, be sure to remember that the graph has a minimum at x=0.5.
Note that this is only one of the infinitely many graphs with the given characteristics. Any graph with two zeros at - 2 and 3, which is decreasing and increasing in the given intervals, will be correct.
Degree and Leading Coefficient of f
From the graph we can tell that f(x) approaches positive infinity as x approaches negative infinity and f(x) approaches positive infinity as x approaches positive infinity.
f(x) → + ∞ as x → - ∞
f(x) → + ∞ as x → + ∞
From the up-up end behavior of the graph we know that our polynomial function has an even degree and a positive leading coefficient.
