POLYNOMIAL FUNCTIONSThe sum or difference of monomial functions.(Exponents are non-negative.)

f(x) = anxn + an-1xn-1 + … + a0

Degree of the polynomial is the degree of the highest degree term.

Leading coefficient is an.

The graph is a smooth and continuous. The domain is all real numbers.

TRANSFORMATIONS OF MONOMIAL FUNCTIONS

Graph these:

y = (x – 4)4 + 2

y = -(x+ 3)5

THE LEADING COEFFICIENT TESTFor any polynomial f(x) = anxn + …

If n is odda) and a > 0, then the graph rises from left to

rightb) and a< 0, then the graph falls from left to

right

If n is evena) and a> 0, then the graph rises on both ends

b) and a< 0, then the graph falls on both ends

NOW TRY THESE… Match the equations to the pictures on

the board.

f(x) = -x3 + 4x

f(x) = x4 – 5x2 + 4

f(x) = x5 – x

Describe the end behavior using limit notation.

REAL ZEROS If f(x) = anxn + …

How many real zeros does f(x) have?At most n.

How many turning points (where it changes from increasing to decreasing vice versa) does f(x) have?

At most n-1.

RELATED IDEAS ABOUT ZEROS x = a is a zero of the function f.

x = a is a solution or root of the equation f(x) = 0.

(x – a) is a factor of f(x).

(a,0) is an x-intercept of the graph of f.

TRY SKETCHING THESE… f(x) = x3 – x2 – 2x

f(x) = -2x4 + 2x2

f(x) = x3 + 3x2 – 4x – 12

f(x) = x4 – 10x2 + 9 (this is in quadratic “form”)

f(x) = (x + 1)3(x – 2)2 (note the effect of the mutiplicity of the roots)

Consider end behavior, y-intercepts, zeros (and their multiplicity).

USING YOUR GRAPHING CALCULATOR - DO #43 ON P. 104

• Go into CATALOG and turn your DIAGNOSTIC ON.

• Make sure your STAT PLOT is on.

• Enter data by going into STAT then EDIT.

• Set an appropriate WINDOW.

• Go into STAT then CALC. Pick an appropriate model.

• Go to Y= and then VARS, STATISTICS, EQ to paste your equation to graph.

• Check the value of R2 to see if your equation is a good fit.

• Some calculators made need to change MODE to CLASSIC.

PART IIReview of 2.1

1. Solve 5(x+1)3/2 = 40

2. Solve 1 + √(x-1) = x

3. Graph y = 7x-2

4. Graph y = x2/3

MORE PRACTICE WITH 2-21. Solve 3x(7x - 2)(x2 – 5)(x2+ 4) = 0

2. Graph f(x) = -2x4 + 16x2 + 18

3. Graph f(x) = x2 (x-2)3 (x+ 1)

NOW TRY THESE1. Write a function with roots 2, 2, -3, and

0.

2. Write a cubic function with zeros only at 1 and 2.

3. Write a quartic equation with roots only at 1 and 2 and no multiple roots.