Download Polynomial Functions

7-1: Polynomial
Functions
-Evaluate Polynomial Functions
-Identify general shapes of graphs and of
polynomial functions
Algebra 2
Review - Simplify
I.
7 13 11 2 8 4 II.
2 1
Terms:
O Polynomial in One Variable –
O Polynomial (ie. NO:
,
, 2 , xy)
O Only ONE variable identified
O Degree of a Polynomial –
O Degree of the largest monomial (greatest
exponent)
O Leading Coefficient –
O Coefficient of the term with the greatest
exponent
O Polynomial Function –
O Function notation f(x); domain vs range
Example 1:
State the degree and leading coefficient of each
polynomial in one variable. If it is not a
polynomial in one variable, explain why. (pg. 346)
a)
b)
c)
Yes, Deg. Of 4, L.C. of 7
NO, more than one variable
! NO, variable in the denominator
d)
Yes, Deg. Of 5, L.C. of -1
Your Turn:
State the degree and leading coefficient of
each polynomial in one variable. If it is not a
polynomial in one variable, explain why.
1)
NO; variable in the radicand
2) " Yes; Deg. Of 4; L.C. of -10
3) NO; variable with a negative exponent
Evaluating Polynomial Functions
Ex. 2:
2: Evaluate each polynomial function
a) Find p(3)
(3) and p(-1) if
p(x)
(x) = -x3 + x2 – x.
Evaluating Polynomial Functions
Ex. 2:
2: Evaluate each polynomial function
b) Find
m(a2) if m(x)
(x) = x3 + 4x2 – 5x.
Evaluating Polynomial Functions
Ex. 2:
2: Evaluate each polynomial function
c) Find
b(2x – 1) – 3b(x) if
b(m) = 2m2 + m – 1.
Evaluating Polynomial
Functions
Your Turn: Evaluate each polynomial function
1) Find p(8a) if # $ % 2) Find r(x + 2) if
& ' $ ' ' ' ' Graphs of Polynomial Functions:
Constant Function
Degree of
0
Linear Function
Degree of
1
Graphs of Polynomial Functions:
Quadratic Function
Degree of
2
Cubic Function
Degree of
3
Graphs of Polynomial Functions:
Quartic Function
Degree of
4
Quintic Function
Degree of
5
Graphs of Polynomial Functions
What do you notice????
Roots/Zeros (real) –
Degree stats MAX. number of real roots
Graphs: Even vs Odd degree –
Even – end in same direction
Odd – end in opposite directions
Will an odd function ALWAYS cross the x-axis?
YES – end in opposite directions
Will an even function ALWAYS cross the x-axis?
NO – end in same direction
Graphs of Polynomial Functions
End Behavior :
Describes the behavior of the graph as it
approaches infinity (∞).
( → ∞%+ → ∞
“the function f of x is approaching positive
infinity as x approaches negative infinity”
(the y-coord. is increasing as the x-coord. is
decreasing)
Ex. 3: For each graph:
1) Describe the end behavior
2) Determine whether the graph represents an odd or even function
3) state the number of real zeros
a)
b)
Ex. 3: For each graph:
1) Describe the end behavior
2) Determine whether the graph represents an odd or even function
3) state the number of real zeros
c)
Your Turn: For each graph:
1) Describe the end behavior
2) Determine whether the graph represents an odd or even function
3) state the number of real zeros
1)
2)
a) ( → ∞%+ → ∞
( → ∞%+ → ∞
b) Even function
c) 2 real zeros
a) ( → ∞%+ → ∞
( → ∞%+ → ∞
b) Odd function
c) 5 real zeros
Homework:
Pg. 350: 17-37 odds, 39-44 all, 57 & 58
Due Thursday