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 $- 2 < x < 3$

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 < x < 3,

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 $0.5 .$ 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) \to + \infty as x \to - \infty f (x) \to + \infty as x \to + \infty

From the up-up end behavior of the graph we know that our polynomial function has an even degree and a positive leading coefficient.

Sketching the Graph

Degree and Leading Coefficient of f

Degree and Leading Coefficient of $f$

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