Algebra II Mr. Gilbert

1
Algebra IIMr. Gilbert

• Chapter 7.1 7.2
• Polynomial Functions
• Graphing Polynomials
• Standard Honors

2
Agenda

• Warm up
• Home Work
• Lesson
• Practice
• Homework

3
Homework Review
4
Communicate Effectively

• Polynomial Function in one variable
• standard form f(x)a0xna1xn-1an-1xan
• a0?0 and a0, a1, an
  are Real
• Degree of the polynomial the largest exponent.
• Leading Coefficient coefficient of the highest
  degree.
• Relative Minimum No other nearby points are
  smaller.
• Relative Maximum No other nearby points are
  larger.

5
Lesson 1 Contents
Example 1 Find Degrees and Leading Coefficients
(5) Example 2 Evaluate a Polynomial Function
(5) Example 3 Functional Values of Variables
(5) Example 4 Graphs of Polynomial Functions (5)
Example 1 Graph a Polynomial Function Example
2 Locate Zeros of a Function Example 3 Maximum
and Minimum Points Example 4 Graph a Polynomial
Model
6
Example 1-1a
Answer This is a polynomial in one variable. The
degree is 3 and the leading coefficient is 7.
7
Example 1-1b
Answer This is not a polynomial in one variable.
It contains two variables, a and b.
Answer This is not a polynomial in one variable.
The term 2c1 is not of the form ancn, where n is
a nonnegative integer.
8
Example 1-1d
Rewrite the expression so the powers of y are in
decreasing order.
Answer This is a polynomial in one variable with
degree of 4 and leading coefficient 1.
9
Example 1-1e
Answer degree 3, leading coefficient 3
Answer This is not a polynomial in one variable.
It contains two variables, x and y.
10
Example 1-1f
Answer This is not a polynomial in one variable.
The term 3a1 is not of the form ancn, where n
is nonnegative.
Answer degree 3, leading coefficient 1
11
Example 1-2a
Find the values of f (4), f (5), and f (6).
12
Example 1-2b
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Example 1-2c
From the information given in Example 2 of your
textbook, you know that the total number of
hexagons for three rings is 19. So, the total
number of hexagons for four rings is 19 18 or
37, five rings is 37 24 or 61, and six rings is
61 30 or 91.
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Example 1-2d
Nature Refer to Example 2 on page 347 of your
textbook. A sketch of the arrangement of hexagons
shows a fourth ring of 18 hexagons, a fifth ring
of 24 hexagons, and a sixth ring of 30
hexagons. Find the total number of hexagons in a
honeycomb with 20 rings.
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Example 1-2e
Answer f (7) 127 f (8) 169 f (9) 217
the total number of hexagons for seven rings is
91 36 or 127, eight rings is 127 42 or 169,
and nine rings is 169 48 or 217. These match
the functional values for r 7, 8, and 9,
respectively.
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Example 1-2f
b. Find the total number of hexagons in a
honeycomb with 30 rings.
Answer 2611
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Example 1-3a
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Example 1-3b
To evaluate b(2x 1), replace m in b(m) with 2x
1.
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Example 1-3c
To evaluate 3b(x), replace m with x in b(m), then
multiply the expression by 3.
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Example 1-3d
Now evaluate b(2x 1) 3b(x).
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Example 1-3e
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Example 1-4a
For the graph, ? describe the end behavior, ?
determine whether it represents an
odd-degree or an even-degree function,
and ? state the number of real zeros.
23
Example 1-4b
For the graph, ? describe the end behavior, ?
determine whether it represents an
odd-degree or an even-degree function,
and ? state the number of real zeros.
24
Example 1-4c
For the graph, ? describe the end behavior, ?
determine whether it represents an
odd-degree or an even-degree function,
and ? state the number of real zeros.
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Example 1-4d
For each graph, a. ? describe the end
behavior, ? determine whether it represents
an odd-degree or an even-degree function,
and ? state the number of real zeros.
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Example 1-4e
For each graph, b. ? describe the end
behavior, ? determine whether it represents
an odd-degree or an even-degree function,
and ? state the number of real zeros.
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Example 1-4f
For each graph, c. ? describe the end
behavior, ? determine whether it represents
an odd-degree or an even-degree function,
and ? state the number of real zeros.
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End of Lesson 1
Graphing
29
Example 2-1a
x f(x)
4 5
3 4
2 3
1 2
0 5
1 0
2 19
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Example 2-1b
This is an odd degree polynomial with a negative
leading coefficient, so f (x) ? ? as x ? ?
and f (x) ? ? as x ? ?. Notice that the graph
intersects the x-axis at 3 points indicating
that there are 3 real zeros.
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Example 2-1c
x f (x)
3 8
2 1
1 2
0 1
1 4
2 17
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Example 2-2a
Make a table of values. Since f (x) is a 4th
degree polynomial function, it will have between
0 and 4 zeros, inclusive.
x f (x)
2 9
1 1
0 1
1 3
2 7
3 19
33
Example 2-2b
Look at the value of f (x) to locate the zeros.
Then use the points to sketch the graph of the
function.
There are zeros between x 2 and 1, x 1
and 0, x 0 and 1, and x 2 and 3.
34
Example 2-2c
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Example 2-3a
Make a table of values and graph the function.
x f (x)
2 19
1 0
0 5
1 2
2 3
3 4
4 5
5 30
zero at x 1
36
Example 2-3b
Answer The value of f (x) at x 0 is greater
than the surrounding points, so it is a relative
maximum. The value of f (x) at x 3 is less than
the surrounding points, so it is a relative
minimum.
x f (x)
2 19
1 0
0 5
1 2
2 3
3 4
4 5
5 30
37
Example 2-3c
Answer The value of f (x) at x 0 is less than
the surrounding points, so it is a relative
minimum. The value of f (x) at x 2 is greater
than the surrounding points, so it is a relative
maximum.
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Example 2-4a
Make a table of values for weeks 0 through 7.
Plot the points and connect with a smooth curve.
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Example 2-4b
n w(n)
0 110
1 109.5
2 108.4
3 107.3
4 106.8
5 107.5
6 110
7 114.9
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Example 2-4c
Describe the turning points of the graph and its
end behavior.
Answer There is a relative minimum at week 4.
For the end behavior, w (n) increases as n
increases.
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Example 2-4d
What trends in the patients weight does the
graph suggest?
Answer The patient lost weight for each of 4
weeks after becoming ill. After 4 weeks, the
patient started to gain weight and continues to
gain weight.
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Example 2-4e
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Example 2-4f
b. Describe the turning points of the graph
and its end behavior.c. What trends in
the amount of rainfall received by the town does
the graph suggest?
Answer There is a relative maximum at Month 2,
or May. For the end behavior, r (m) decreases as
m increases.
Answer The rainfall increased for two months
following March. After two months, the amount of
rainfall decreased for the next five months and
continues to decrease.
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Homework
See Syllabus 7.1 7.2