Polynomial Functions and their Graphs

Section 3.1

General Shape of Polynomial Graphs

The graph of polynomials are smooth, The graph of polynomials are smooth, unbroken lines or curves, with no sharp unbroken lines or curves, with no sharp corners or cusps corners or cusps (see p. 251).

Every Polynomial function is defined and continuous for all real numbers.

Review

General polynomial formula

a0, a1, … ,an are constant coefficients n is the degree of the polynomial Standard form is for descending powers of x aannxxn

n is said to be the “leading termleading term”

11 1 0( ) ...n n

n nP x a x a x a x a

Family of Polynomials

Constant polynomial functions f(x) = a

Linear polynomial functions f(x) = mx + b

Quadratic polynomial functions f(x) = ax2 + bx + c

Family of Polynomials

Cubic polynomial functions f(x) = a x3 + b x2 + c x + d 3rd degree polynomial

Quartic polynomial functions f(x) = a x4 + b x3 + c x2+ d x + e 4th degree polynomial

Polynomial “End BehaviorEnd Behavior”

Consider what happens when x gets very large in positive and negative direction Called “end behavior” Also “long-run” behavior

Basically, the leading term Basically, the leading term aannxxnn

dominates the shape of graphdominates the shape of graph There are 4 possible scenarios:

End Behavior

Discuss end behavior for the following graphs:

4 3( ) 2 5 4 7P x x x x

5 4 2( ) 2 6 10P x x x x 5 3( ) 3 5 2P x x x x

Compare Graph Behavior

Consider the following graphs: f(x) = x4 - 4x3 + 16x - 16 g(x) = x4 - 4x3 - 4x2 +16x h(x) = x4 + x3 - 8x2 - 12x

Graph these on the window

-8 < x < 8       and      0 < y < 4000 Decide how these functions are alike or

different, based on the view of this graph

Compare Graph Behavior

From this view, they appear very similar

Compare “Short Run” Behavior

Now Change the window to be-5 < x < 5   and   -35 < y < 15

How do the functions appear to be different from this view?

Compare Short Run Behavior

Differences? Real zeros Local extrema Complex zeros

Note: The standard form of the polynomials does not give any clues as to this short run behavior of the polynomials:

Using Zeros to Graph PolynomialsUsing Zeros to Graph Polynomials

Consider the following polynomial: p(x) = (x - 2)(2x + 3)(x + 5)

What will the zeros be for this polynomial? x = 2 x = -3/2 x = -5

How do you know? Zero-Factor Property: If a*b = 0 then, we know

that either a = 0 or b = 0 (or both)

Guidelines to Graphing

ZerosZeros Test Points (like table of signs)Test Points (like table of signs) End BehaviorEnd Behavior Graph (a smooth curve through all Graph (a smooth curve through all

known points)known points)

Intermediate Value Theorem

If P is a polynomial function and P(a) and P(b) have opposite signs, then there is at least one value c between a and b for which P(c) = 0.

Theorem

Local Extrema of Polynomial Functions: A polynomial function of degree A polynomial function of degree nn has at most has at most nn - 1 local extrema. - 1 local extrema.

Local Extrema (turning points)

Local Extrema – a point (x,y) on the graph where the graph changes from increasing to decreasing or vice-versa.

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