Chapter 3.4

Polynomial Functions

y

x

Graphy = x3

y = x5

y = x4

y = x6

As with quadratic functions, the value of a in f(x) = axn determines the width of the graph.

When |a| > 1, the graph is stretched vertically, making it narrower

while when 0 < |a| < 1, the graph is shrunk or compressed vertically, so the graph is broader.

y

x

Graphy = x2

y = 2x2

y = 1/2x3

The graph of f(x) = -axn is reflected across the x-axis compared to the graph of f(x) = -axn

y

x

Graphy = x2

y = -x2

Compared with the graph of f(x) = axn

the graph of f(x) = axn + k is translated (shifted) k units up if k > 0.

Also, when compared with the graph of f(x) = axn

the graph of f(x) = a(x-h)n is translated h units to the right if k > 0 and |h| units to the left if k < 0.

The graph of f(x) = a(x-h)n + k shows a combination of these translations. The effects here are the same as those we saw earlier with quadratic functions.

y

x

Graphy = x5 - 2

y

x

Graphy = (x+1)6

y

x

Graphy = -2(x-1)3 +3

The domain of every polynomial function is the set of all real numbers; thus polynomial functions are continuous on the interval (-∞, ∞).

The range of a polynomial function of odd degree is also the set of all real numbers.

Typical graphs of polynomial functions of odd degree are shown in Figure 22. These graphs suggest that for every polynomial function f of odd degree there is at least one real value of x that makes f(x) = 0.

The zeros are the x-intercepts of the graph.

A polynomial function of even degree has range of the form (-∞, k] or [k, ∞) for some real number k. Figure 23 shows two typical graphs of polynomial functions of even degree.

22 )2(1)(x y :tyMultiplici x

axis.- x thecrosses

graph theodd, isexponent theIf

axis.- x thetouches

graph theeven, isexponent theIf

The end behavior of a polynomial graph is determined by the dominating term, that is, the term of greatest degree.

A polynomial of the formf(x) = anxn + an-1xn + . . . + a0 has the same endbehavior as f(x) = anxn.

For instancef(x) = 2x3 – 8x2 + 9 has the same end behavior as f(x) = 2x3. It is large and positive for large positive values of x and large and negative for large negative values of x. with large absolute value.

The arrows at the ends of the graph look like those of the first graph in Figure 22; the right arrow points up and the left arrow points down.

The first graph in Figure 22 shows that as x takes on larger and larger positive values, y does also. This is symbolized

,y , xas

For the same graph, as x takes on negative values of larger and larger absolute value, y does also.

.y , xas

For the middle graph in Figure 22 we have

y , xas

y , xas

Graphing TechniquesA comprehensive graph of a polynomial funciton

will show the following characteristics.

1. all x-intercepts2. the y-intercept3. all turning points4. enough of the domain to show the end

behavior.

We emphasize the important relationships among

the following concepts.

1. the x-intercepts of the graph of y = f(x)2. the zeros of the function f3. the solutions of the equation f(x) = 04. the factors of f(x)

Caution:Be careful how you interpret the intermediate value theorem.

If f(a) and f(b) are not opposite in sign, it does not necessarily mean that there is no zero between a and b.

For example, in Figure 30, f(a) and f(b) are both negative, but -3 and -1, which are between a and b, are zeros of f(x).

Use synthetic division to show that f(x) = x3 -2x2 – x + 1 has a real zero between 2 and 3.

Use synthetic division to find f(2) and f(3)

Use synthetic division to show that f(x) = x3 -2x2 – x + 1 has a real zero between 2 and 3.

Use synthetic division to find f(2) and f(3)

Since f(2) is negative and f(3) is positive, by the intermediate value theorem there must be a real zero between 2 and 3.

Show that the real zeros of f(x) = 2x4 -5x3 + 3x + 1satisfy the following conditions.

(a) No real zero is greater than 3

Show that the real zeros of f(x) = 2x4 -5x3 + 3x + 1satisfy the following conditions.

(b) No real zero is less than -1