Polynomial Functions

The degree of a polynomial function helps us to determine the number of $x$-intercepts and the number of turning points. A polynomial function of $n \mathrm{th}$ degree is the product of $n$ factors, so it will have at most $n$ roots or zeros, or $x$-intercepts. The graph of the polynomial function of degree $n$ must have at most $n–1$ turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.

A continuous function has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.

INTERCEPTS AND TURNING POINTS OF POLYNOMIALS

A polynomial of degree $n$ will have, at most, $n$ $x$-intercepts and $n−1$ turning points.

EXAMPLE 10

Determining the Number of Intercepts and Turning Points of a Polynomial

Without graphing the function, determine the local behavior of the function by finding the maximum number of x-intercepts and turning points for $f(x)=-3 x^{10}+4 x^{7}-x^{4}+2 x^{3}$.

Solution

The polynomial has a degree of $10$, so there are at most $10$ $x$-intercepts and at most $9$ turning points.

TRY IT #7

Without graphing the function, determine the maximum number of x-intercepts and turning points for $f(x)=108-13 x^{9}-8 x^{4}+14 x^{12}+2 x^{3}$.

EXAMPLE 11

Drawing Conclusions about a Polynomial Function from the Graph

What can we conclude about the polynomial represented by the graph shown in Figure 12 based on its intercepts and turning points?

Figure 12

Solution

The end behavior of the graph tells us this is the graph of an even-degree polynomial. See Figure 13.

Figure 13

The graph has $2$ $x$-intercepts, suggesting a degree of $2$ or greater, and $3$ turning points, suggesting a degree of $4$ or greater. Based on this, it would be reasonable to conclude that the degree is even and at least $4$.

TRY IT #8

What can we conclude about the polynomial represented by the graph shown in Figure 14 based on its intercepts and turning points?

Figure 14

EXAMPLE 12

Drawing Conclusions about a Polynomial Function from the Factors

Given the function $f(x)=-4 x(x+3)(x-4)$, determine the local behavior.

Solution

The $y$-intercept is found by evaluating $f(0)$.

$\begin{aligned} f(0) &=-4(0)(0+3)(0-4)\\ &=0 \end{aligned}$

The $y$-intercept is $(0,0)$.

The $x$-intercepts are found by determining the zeros of the function.

$0=-4 x(x+3)(x-4)$

$x=0 \quad \text { or } \quad x+3=0 \quad \text { or } \quad x-4=0$

$x=0 \quad \text { or } \quad x=-3 \quad \text { or } \quad x=4$

The $x$-intercepts are $(0,0)$, $(–3,0)$, and $(4,0)$.

The degree is $3$ so the graph has at most $2$ turning points.

TRY IT #9

Given the function $f(x) \equiv 0.2(x-2)(x+1)(x-5)$, determine the local behavior.