3 1 Polynomial Functions and their Graphs fx

• The graph of a polynomial function is always a smooth curve –

End Behavior & the Leading Coefficient • Tells what happens as x becomes large

End behavior is determined by the term that contains the highest power of x.

EX Determine the end behavior: • f(x) = -3 x 3 + 20 x

Using Zeros to Graph If P is a polynomial and c is a real

Find the zeros by factoring: • P(x) = x 2 + x - 6

If you have positive and negative yvalues, your polynomial has to have at least

Table must include: • Zeros • A point in between each zero • Y-intercept

P(x) = (x + 2)(x – 1)(x – 3) • Find the zeros and

Multiplicity m is the exponent Passes through Bounces off

P(x) = (x + 2)(x – 1)(x – 3)2 • Find the zeros and

PHONE NUMBER • Key in the 1 st three digits of your phone number

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

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

Slides: 21

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3. 1 Polynomial Functions and their Graphs

f(x) = 5 3 x + 4 6 x – 3 2 x + 7 x - 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 = x 4 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

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) = -3 x 3 + 20 x 2 + 60 x + 2 • f(x) = -7 x 4 + 5 x 3 + 4 x - 7

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) = x 2 + x - 6

If you have positive and negative yvalues, 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 0 1 0 3 0

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 0 1 0 3 0

PHONE NUMBER • Key in the 1 st 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 # AGAIN • Subtract 250 • Divide by 2 • SEE YOUR PHONE NUMBER ON YOUR CALCULATOR? ? ?

P(x) = 3 x 4 – 5 x 3 – 12 x 2 • Find the zeros, y-int, and graph. X F(x) -4/3 0 0 0 3 0

P(x) = x 3 + 3 x 2 – 9 x - 27 • Find the zeros, y-int, and graph. X F(x) -3 0

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

P(x) = 3 x 4 – 5 x 3 – 12 x 2 • Find the zeros, y-int, and graph. X F(x) -4/3 0 0 0 3 0

P(x) = (x + 2)(x – 1)(x – 3) • Find the zeros and graph. X F(x) -2 0 1 0 3 0

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