Polynomials have the property that the sum, difference and product of polynomialsalways produce another polynomial.

In this chapter we will be studying rational functions and algebraic functions.

Ex: are both

algebraic functions; g(x) is a rational function since it is the quotient of twopolynomials.

2352

)(12)(4

234

xxxxx

xgandxxxf

A polynomial of degree n, where n is a non-negative integer, has the form

01

1

1 ...)( axaxaxaxP n

n

n

n

naaa ..., 10are all constants where a ≠ 0.

Polynomial Degree Leading

Coefficient

Constant

Term

123 246 xxxx 6 -3 1

17 0 17 17

522 23 xxx 3 2 -5

10x 10 1 0

Recall: We have seen that the graphs of polynomials of degree 0 or 1 are lines and the graph of a polynomial of degree 2 is a parabola.

Ex 1: Use learned techniques to graph 1)2(3 3 xy

1st: we start with the graph of the parent function.3xy

2nd: We shift the graph to the right 2 units.

3rd: We multiply by -3 which will reflect the graph over the x-axis.

4th: Shift the graph 1 unit up.

Graph together on board.

Ex 2: Sketch the graph of xxxf 4)( 3

1st: Notice the graph has 3 x-intercepts

2nd: As x becomes large, is much larger than -4x. So, the graph of f(x) will approach , and its graph will be similar.

3x3xy

Graph together on board.

Note: The graph has origin symmetry, therefore, it is an odd function.

End Behavior of a Graph: The behavior of the y-values of points on the curve for large x-values, as well as x-values that are negative but with large magnitude.

Note: the end behavior of any polynomial function of degree n depends solely on the leading coefficient, an.

All polynomials of degree n≥1 go to as x goes to .

Examples of end behavior p. 128 top

Zero of a function: Where f(c)= 0; the x-intercepts.

Every polynomial function has the property called continuity.

Continuity means the graph has no breaks or interruptions, this is the essence ofthe Intermediate Value Theorem.

Intermediate Value Theorem: if f is continuous on [a, b] and if k is any numberbetween f(a) and f(b), then some number c between (a, b) exists with f(c)= k

y

x

f(b) kf(a)

a c b

k = f(c)

Note: The Intermediate ValueTheorem tells us that continuousfunctions do not skip over any valuesin the range.

The graph of a polynomial betweensuccessive zeros is either always positive or always negative, meaninglying above or below the x-axis.

Ex 3: Graph xxxxf 2)( 23

Solution: 1st: Find the zeros of the function.

Check to see if the terms have a common factor.

)1)(2()2()( 2 xxxxxxxf

x = 0, x - 2 = 0, x + 1 = 0 x = 2 x = -1

Zeros = x- intercepts = x = -1, 0, 2

2nd: We must now determine whether the graph between successive zeros lies above or below the x-axis

Sign graph~

xx – 2x + 1x(x-2)(x+1)

-1 0 1 2

----------------------0++++++++++++++++----------------------------------------0++++++------------- 0+++++++++++++++++++++--------------0+++ 0----------------0++++++

below above below above

“ – “ means below the x-axis

“+” means above the x-axis

We now know the zeros and on what intervals the graph will lie above and belowthe x-axis.

To determine the End Behavior of the graph we consider f(x) in its original form.

xxxxf 2)( 23

f(x) has an odd degree of 3 and a leading coefficient of 1, therefore, the polynomial behaves like the parent function, in large magnitude.,3xy

Now, with all of our info together we can sketch the graph.

y

x

localmaximum

localminimum

Local maximums and local minimumsare called local extrema

Think Pair Share:

Ex 4: Graph 234 23)( xxxxf

Solution: )1)(2()23()( 222 xxxxxxxf

Zeros: x = 0, 1, 2

x2

x - 2x – 1x2(x – 2)(x – 1)

-1 0 1 2

++++++++++++0++++++++++++++++------------------------------------0+++++++-----------------------------0++++++++++++++++++++++++0+++ 0----- 0++++++++ above above below above

Since the degree is 4 and the leading coefficient is positive it will behave like y = x4