SB CH 2 answers.notebook
November 05, 2013
Warm Up
Oct 8­10:36 AM
Oct 5­2:22 PM
Linear Function Qualities
Oct 8­9:22 AM
Oct 8­9:19 AM
Quadratic Function Qualities
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Oct 8­9:25 AM
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SB CH 2 answers.notebook
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November 05, 2013
Oct 8­9:25 AM
Given vertex (-1,4) and point (1,2), write the equation for the
quadratic function.
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Oct 8­9:23 AM
Vertical Free Fall Motion
Homework:
Page 169
#1-12 odds, 13-18, 23-44 odds
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Oct 8­9:26 AM
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SB CH 2 answers.notebook
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Homework: Page 170 # 45-48, 51, 54-59, 61-63, 69, 70
and finish homework from pg 169.
Example:
Given: f(x) = x2+3
Find the average rate of change from x = 3 to x = 6.
Oct 8­9:22 AM
Oct 8­10:12 AM
Warm-Up
Oct 8­10:42 AM
Oct 8­10:44 AM
Example 1:
Example 2: State whether the following functions are power functions.
If yes, give the power and leading coefficient.
a. f(x) = 16x2/3
b. f(x) = 5(2x)
c. f(x) = 8ax6
Oct 8­10:46 AM
Oct 8­10:44 AM
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SB CH 2 answers.notebook
November 05, 2013
Example: State whether the following are monomial functions or not.
If yes, state the degree and leading coefficient. If not, say why not.
a. f(x) = 4 x8
3
b. y = 5x-3
c. -7(6x)
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Oct 8­10:45 AM
Examples: Write the statement as a power function equation. USe k
for the constant of variation if one is not given.
a. The volume of a circular cylinder with fixed height is proportional
to the square of its radius r.
Homework: Page 182
#1-22 odds, 37-42 all, 53-57 odd, 58-62 all
b. Charles Law states that the volume V of an enclosed ideal gas at a
constant pressure varies directly as the absolute temperature T.
c. Distance d varies inversely with time t and has a constant of
variation of k = velocity v.
d. The speed p of a falling object that has been dropped from rest
varies as the square root of the distance d traveled, with a constant
of variation k = √2g
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Oct 8­10:49 AM
Section 2.3 Polynomial Functions of Higher Degree with Modeling
Recall : the degree of a polynomial is the highest power. The leading
coefficient is the number attached to the highest power term.
Oct 17­8:44 PM
where a3 is the leading coefficient, if:
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SB CH 2 answers.notebook
November 05, 2013
where a4 is the leading coefficient, if:
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Oct 17­8:48 PM
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Oct 17­8:49 PM
aka: end behavior is
determined by the
leading coefficient and
degree of the polynomial.
Make chart of
even vs odd to
know end behavior
Better Explained as:
If there is a power on the outside of the linear factor for the zero,
then the zero has multiplicity.
If a zero has odd multiplicity, the graph of the function crosses the
x-axis at that zero.
If a zero has even multiplicity, the graph of the function touches the
x-axis at that zero but does not go through the x-axis.
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Oct 17­9:00 PM
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SB CH 2 answers.notebook
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Example: Graph the function in a viewing window that shows all of
its extrema and x-ints. Describe end behavior with limits.
f(x) = (2x-3)(4-x)(x+1)
w/ y-int of 8
f(x) = -3x4 - 5x3 - 17x2 + 14x + 41
plot y­int and zeroes first, then figure out end behavior, then look at multiplicities with zeroes to connect inside in graph together.
Oct 17­8:59 PM
Oct 17­9:12 PM
Homework:
Page 193
#1-7 odds, 9-12, 33-42 odd, 67, 69-73
Hint: when matching an equation with a graph without a
calculator- look at leading coefficient and degree of the
polynomial first.
**In practice, the IVT is used in combination with our other
math knowledge to explain or prove statements.
See Ex 7 page 190
Oct 17­8:58 PM
Oct 17­9:24 PM
2.4 Real Zeros of Polynomial Functions
Example: Use long division to find the quotient and remainder
when 2x4 - x3 - 2 is divided by 2x2 + x + 1. Write your answer in
polynomial and fractional form.
Example: Use syntheic division to divide 2x3 - 3x2 - 5x - 12 by x - 3.
3
2 -3
-5
-12
2
= 2x3 ­ 3x2 ­ 5x ­ 12
fractional form
2x4 - x3 - 2
= x2 ­ x + x+2
2x2 + x + 1
2x2 + x +1
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polynomial form
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SB CH 2 answers.notebook
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Remainder Theorem: If a polynomial f(x) is divided by x - k then
the remainder is r = f(k).
Factor Theorem: A polynomial function f(x) has a factor x-k if and
only if f(k) = 0.
Example: Find the remainder when f(x) = 3x2 + 7x - 20
Example: Show that x+4 is a factor of 3x2 + 7x - 20
is divided by: a. x­2 b. x + 1 c. x + 4
f(­4) = 3(­4)2 + 7(­4) ­ 20
x+4 3x2 + 7x - 20
48 ­ 28 ­20 = 0
­­­­OR­­­­ ­­­­OR­­­­ f(2)= 3(2)2 + 7(2) ­ 20
f (­1) = 3(­1)2 + 7(­1) ­20
12 + 14 ­ 20 = 6
3 ­ 7 ­ 20 = ­24
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Oct 18­10:12 PM
the one zero we can definitely find in our table
Note: Use synthetic
division to confirm
potential zeroes.
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this yields: 2x2 - 4 (factoring out a 2 gives):
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See page 202 with Upper & Lower bounds and Ex 6 on page 203
Homework: Page 205 # 1-23 odd
Section 2.4
(Part 2)
p. 205 25 ­ 35 odd
49 ­ 55 odd
57 ­ 60, 65 ­ 68
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Oct 19­9:27 AM
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SB CH 2 answers.notebook
Section 2.5
Complex Zeroes & The Fundamental Theorem of Algebra
November 05, 2013
Example: Write the polynomial function in standard form and
identify the zeroes and x-intercepts.
a. f(x)= (x - 2i)(x + 2i)
Fundamental Theorem of Algebra: A polynomial of degree n has n
complex zeroes (real & nonreal).
b. f(x) = (x - 3)(x - 3)(x - i)(x + i)
Oct 19­9:16 AM
Example: Write a polynomial function of minimum degree in standard
form with real coefficients whose zeroes include those listed.
Oct 19­9:22 AM
Example: Write a polynomial function of minimum degree
in standard form with real coefficients whose zeroes and
their multiplicities include those listed.
Zeroes are 2, -3, and 1 + i .
Given: -1 with multiplicity 3 and 3 with multiplicity 1
Oct 19­9:23 AM
Oct 19­9:29 AM
Homework: Page 215 #2, 9, 12, 15, 17-20, 27, 29, 31, 34, 37, 39
Oct 19­9:27 AM
Oct 19­9:27 AM
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SB CH 2 answers.notebook
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Section 2.6 Graphs of Rational Functions
Oct 19­10:57 AM
Oct 19­11:04 AM
Graphs of Rational Functions
Example: Given the graph below, evaluate the limits.
a. lim f(x)
x
-1+
b. lim f(x)
x
-
y-int: occur when the value of f(0) is defined
(remember x = 0 gives the y-int)
-1
c. lim f(x)
x
x-int: occur when the numerator = 0 (but it cannot also
make the den = 0)
-∞
d. lim f(x)
Asymptotes Information
V. A. occur when the den = 0.
H.A. occur when the
x +∞
1: (degree of the num) = (deg of den)
-in this case, the H.A. is "y = ratio of the leading
coefficients of num. & den."
2: (degree of den.) is larger than (degree of num.)
-in this case, the H.A. is y = 0
Oct 20­10:45 AM
Oct 19­11:05 AM
Example:
x-int:
y-int:
V.A.
H.A.
In Summary:
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SB CH 2 answers.notebook
November 05, 2013
**You do NOT have to graph the functions as stated in the
directions- but do need to find all the information it is asking for.
p. 225 1­10 mod 3, 11­14 23, 24, 27, 28
31 ­ 36
41, 42, 53
65 ­ 68
Oct 20­10:42 AM
Oct 21­11:00 AM
Solving Equations in One Variable
3 creates a "0" on the bottom of the first fraction, and cant divide by 0 in math.
Oct 19­10:58 AM
Oct 21­11:00 AM
Homework: Page 232 #1-18, 23-30 mod 3
p. 232 31, 32, 33, 35
37, 42
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Oct 21­11:03 AM
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SB CH 2 answers.notebook
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Section 2.8 Solving Inequalities in One Variable
factor
Oct 19­10:58 AM
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Oct 21­11:13 AM
*Make a sign chart
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SB CH 2 answers.notebook
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Absolute value ex from pg 240
Example: Solving an inequality involving absolute value
p. 242 7 ­ 23 odd
56 ­ 61
Solve: x - 2 ≤ 0
x+3
because abs. value of x + 3 is in the denominator, f(x) is
undefined at x = -3. The only zero is 2.
Section 2.8
(Part 2)
p. 242 33 ­ 50 mod 3
-3
2
Solution: (-∞,-3) ∪ (-3, 2]
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