Post on 19-Jan-2016
Polynomial Functions
Advanced MathChapter 3
Quadratic Functions and Models
Advanced MathSection 3.1
Advanced Math 3.1 - 3.3 3
Quadratic function
• Polynomial function of degree 2
2f x ax bx c
Advanced Math 3.1 - 3.3 4
Parabola
• “u”-shaped graph of a quadratic function
• May open up or down
Advanced Math 3.1 - 3.3 5
Axis of symmetry
• Vertical line through the center of a parabola
• Vertex: where the axis intersects the parabola
Math Composer 1. 1. 5http: / /www. mathcomposer. com
vertex
-8 -7 -6 -5 -4 -3 -2 -1 1 2
-3
-2
-1
1
2
3
4
5
6
7
8
x
y
Advanced Math 3.1 - 3.3 6
Standard Form
• Convenient for sketching a parabola because it identifies the vertex as (h, k).
• If a > 0, the parabola opens up• If a < 0, the parabola opens down
2
0
f x a x h k
a
Advanced Math 3.1 - 3.3 7
Graphing a parabola in standard form
• Write the quadratic function in standard form by completing the square.
• Use standard form to find the vertex and whether it opens up or down
22 1
example
f x x x
Advanced Math 3.1 - 3.3 8
Example
• Sketch the graph of the quadratic function
2 4 1f x x x
Advanced Math 3.1 - 3.3 9
Writing the equation of a parabola
• Substitute for h and k in standard form
• Use a given point for x and f(x) to find a
example
vertex: 4,-1
point 2,3
Advanced Math 3.1 - 3.3 10
Example
• Write the standard form of the equation of the parabola that has a vertex at (2,3) and goes through the point (0,2)
Advanced Math 3.1 - 3.3 11
Finding a maximum or a minimum
• Locate the vertex
• If a > 0, vertex is a minimum (opens up)
• If a < 0, vertex is a maximum (opens down)
2:
: ,2 2
function f x ax bx c
b bvertex f
a a
Advanced Math 3.1 - 3.3 12
Example
• The profit P (in dollars) for a company that produces antivirus and system utilities software is given below, where x is the number of units sold. What sales level will yield a maximum profit?
20.0002 140 250000P x x
Polynomial Functions of Higher Degree
Advanced MathSection 3.2
Advanced Math 3.1 - 3.3 14
Polynomial functions
• Are continuous– No breaks– Not piecewise
• Have only smooth rounded curves– No sharp points
• This section will help you make reasonably accurate sketches of polynomial functions by hand.
Advanced Math 3.1 - 3.3 15
Power functions
• n is an integer greater than 0• If n is even, the graph is similar to
f(x)=x2
• If n is odd, the graph is similar to f(x)=x3
• The greater n is, the skinner the graph is and the flatter the it is near the origin.
nf x x
Advanced Math 3.1 - 3.3 16
Compare
2
4
6
f x
x
x
f
x
x
f x
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-4 -3 -2 -1 1 2 3 4
-1
1
2
3
4
5
6
7
8
9
10
x
y
Advanced Math 3.1 - 3.3 17
Compare
3
5
7
f x
x
x
f
x
x
f x
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-3 -2 -1 1 2 3
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
x
y
Advanced Math 3.1 - 3.3 18
Examples
• Sketch the graphs of:
5
5
5
5
1
1
11
2
y x
y x
y x
y x
Advanced Math 3.1 - 3.3 19
The Leading Coefficient Test
• If n (the degree) is odd• The left and right go opposite directions• A positive leading coefficient means the
graph falls to the left and rises to the right– As x becomes more positive, the graph goes
up• A negative leading coefficient means
the graph rises to the left and falls to the right– As x becomes more negative, the graph
goes up
Advanced Math 3.1 - 3.3 20
The Leading Coefficient Test
• If n (the degree) is even• The left and right go the same direction• A positive leading coefficient means the
graph rises to the left and rises to the right– The graph opens up
• A negative leading coefficient means the graph falls to the left and falls to the right– The graph opens down
Advanced Math 3.1 - 3.3 21
The Leading Coefficient test
• Does not tell you how many ups and downs there are in between
• See the Exploration on page 277
Advanced Math 3.1 - 3.3 22
Zeros of polynomial functions
• For a polynomial function of degree n
• There are at most n real zeros• There are at most n – 1 turning
points (where the graph switches between increasing and decreasing).
• There may be fewer of either
Advanced Math 3.1 - 3.3 23
Finding zeros
• Factor whenever possible• Check graphically
2
3 2
5 3
6 9
20
2
examples
h t t t
f x x x x
g x x x x
Advanced Math 3.1 - 3.3 24
Repeated zeros
A factor , 1, yields a of .
1. If is odd, the graph the x-axis at .
2. If is even, the graph the x-axis (but doesn't cross it)
at
kx a k x a k
k crosses x a
k touches
x a
repeated zero multiplicity
Advanced Math 3.1 - 3.3 25
Standard form
• For a polynomial greater than degree 2– Terms are in descending order of
exponents from left to right
Advanced Math 3.1 - 3.3 26
Graphing polynomial functions
1. Write in standard form
2. Apply leading coefficient test
3. Find the zeros4. Plot a few
additional points5. Connect the
points with smooth curves.
2 33
example
f x x x Math Composer 1. 1. 5http: / /www. mathcomposer. com
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
Advanced Math 3.1 - 3.3 27
Examples
2 35f x x x
2 212 2
4f x x x
Advanced Math 3.1 - 3.3 28
The Intermediate Value Theorem
• See page 282• Helps locate real zeros• Find one x value at which the
function is positive and another x value at which the function is negative
• Since the function is continuous, there must be a real zero between these two values
• Use the table on a calculator to get closer to the zero and approximate it
Advanced Math 3.1 - 3.3 29
Examples
• Use the intermediate value theorem and the table feature to approximate the real zeros of the functions. Use the zero or root feature to verify.
4 212
4f x x x
2 212 3 5
5f x x x
Polynomial and Synthetic Division
Advanced MathSection 3.3
Advanced Math 3.1 - 3.3 31
Long division of polynomials
• Write the dividend in standard form
• Divide– Divide each term by the leading term
of the divisor
217 5 12 4
example
x x x
Advanced Math 3.1 - 3.3 32
Examples
3 24 7 11 5 4 5x x x x
3 26 16 17 6 3 2x x x x
Advanced Math 3.1 - 3.3 33
Checking your answer
• Graph both the original division problem and your answer
• The graphs should match exactly
Advanced Math 3.1 - 3.3 34
Remainders
• Write remainder as a fraction with the divisor on the bottom
• Examples:
3 24 3 12 3x x x x
3 29 1x x
4 2 23 1 2 3x x x x
Advanced Math 3.1 - 3.3 35
Division Algorithm
• Get rid of the fraction in the remainder by multipling both sides by the denominator.
4 22
2 2
3 1 2 112 4
2 3 2 3
x x xx x
x x x x
dividend divisor quotient remainder
f x d x q x r x
Advanced Math 3.1 - 3.3 36
Synthetic division
• The shortcut• Works with divisors of the form x –
k , where k is a constant• Remember that x + k = x – (– k)
Advanced Math 3.1 - 3.3 37
Synthetic division
• Use an L-shaped division sign with k on the outside and the coefficients of the dividend on the inside
• Leave space below the dividend
• Add the vertical columns, then multiply diagonally by k
24 10 21
5
example
x x
x
Advanced Math 3.1 - 3.3 38
Examples
3 23 17 15 25 5x x x x
3 75 250 10x x x
2 35 3 2 1x x x x
Advanced Math 3.1 - 3.3 39
Remainder Theorem
• If a polynomial f(x) is divided by x – k , then the remainder is r = f(k)
• The remainder is the value of the function evaluated at k
Advanced Math 3.1 - 3.3 40
Examples
• Write the function in the form f(x) = (x – k)q(x) + r for the given value of k , and demonstrate that f(k) = r
3 25 11 8
2
f x x x x
k
3 22 5 4
5
f x x x x
k
Advanced Math 3.1 - 3.3 41
If the remainder is zero…
• (x – k) is a factor of the dividend• (k, 0) is an x-intercept of the graph
Advanced Math 3.1 - 3.3 42
Example
• Show that (x + 3) and (x – 2) are factors of f(x) = 3x3 + 2x2 – 19x + 6.
• Write the complete factorization of the function
• List all real zeros of the function