3.1 Polynomial Functions and their Graphs

f(x) = 3x5 + 6x4 – 2x3 + 7x - 6

The graph of a polynomial function is always a smooth curve – no breaks, holes, or corners. Recall section 2.4 about stretches, shifts, etc. EX: Graph y = x4 and y = (x – 2)4

Polynomial?

End Behavior & the Leading Coefficient Tells what happens as x becomes large in the positive or negative direction. x gets bigger x gets smaller

Always opposite Always the same End behavior is determined by the term that contains the highest power of x. (leading coefficient) Always opposite Always the same

EX Determine the end behavior: f(x) = -3x3 + 20x2 + 60x + 2 f(x) = -7x4 + 5x3 + 4x - 7

Find factors Find zeros Using Zeros to Graph If P is a polynomial and c is a real number, then the following are equivalent: ‘c’ is a zero if P(c) = 0 X = c is an x-intercept of the graph of P X = c is a solution of the equation P(x) = 0 X – c is a factor of P(x) Find factors Find zeros

Find the zeros by factoring: P(x) = x2 + x - 6

If you have positive and negative y-values, your polynomial has to have at least one zero.

Table must include: Zeros A point in between each zero Y-intercept Need to know end behavior

P(x) = (x + 2)(x – 1)(x – 3) Find the zeros and graph. X F(x) -2 1 3

Multiplicity m is the exponent Passes through Bounces off

P(x) = (x + 2)(x – 1)(x – 3)2 Find the zeros and graph. X F(x) -2 1 3

Phone number Key in the 1st three digits of your phone number (not the area code) Multiply by 80 Add 1 Multiply by 250 Add the last four digits of your phone # Add the last four digits of your phone # AGAIN Subtract 250 Divide by 2 SEE YOUR PHONE NUMBER ON YOUR CALCULATOR???

P(x) = 3x4 – 5x3 – 12x2 Find the zeros, y-int, and graph. -4/3 3 X F(x) -4/3 3

P(x) = x3 + 3x2 – 9x - 27 Find the zeros, y-int, and graph. -3 3 X F(x) -3 3

The number of local extrema must be less than the degree.

P(x) = 3x4 – 5x3 – 12x2 Find the zeros, y-int, and graph. -4/3 3 X F(x) -4/3 3

P(x) = (x + 2)(x – 1)(x – 3) Find the zeros and graph. X F(x) -2 1 3

Homework pg 262 #1, 3, 5 - 10, 13, 15, 16, 21, 25-45 odd