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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