Download Math Analysis Honors – MATH Sheets M = Modeling A = Again T

Math Analysis Honors – MATH Sheets
M = Modeling
A = Again
T = Today’s Topic
H = Homework
“M” is the sample problems for the day
“A” are review problems - “A” questions are ALWAYS done first!!
“T” are the objectives of the day
“H” are homework problems
#15 Monday 9/22________________________________
M
Ex. 1 Given f x   ax 2  bx  c , find the x-coordinate of the vertex by completing the square and
converting the standard form equation into vertex form.
Ex. 2 Locate the vertex and axis of symmetry of the parabola defined by f x   3x 2  6x 1 . Graph f x .
Ex. 3 Graph f x   x 2  6x  9 . Determine where f is increasing and where it is decreasing.
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T
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1
x  2  5 , list the transformations that occur in order to change f x  into g x , if f x   x
2
Graph a Quadratic Function in Standard Form using its Vertex and Axis of Symmetry
Worksheet 13
Given g x   
#16 Tuesday 9/23_______________________________
M
Ex. 1 Determine whether the quadratic function, f x   x 2  4x  5 , has a maximum or minimum. Find the
maximum or minimum value.
Ex. 2 The marketing department at Texas Instruments has found that, when certain calculators are sold at a price p
dollars per unit, the revenue R (in dollars) as a function of the price p is R  p   150 p2  21, 000 p . What unit
price should be established to maximize revenue? If this price is charged, what is the maximum revenue?
Ex. 3 A farmer has 2000 yards of fence to enclose a rectangular field. What are the dimensions of the rectangle
that encloses the most area?
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Solve the quadratic equation by any method: 3x2  4x  2  0
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Use the Max or Min Value of a Quadratic Function to Solve Applied Problems
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Worksheet 14
#17 Wednesday 9/24 ________________
M
Ex. 1 A rock thrown vertically upward from the surface of the moon at a velocity of 24 m/sec reaches a height of
h t   24t  0.8t 2 meters in t seconds.
(a) How long did it take the rock to reach its highest point?
(b) How high did the rock go?
(c) When did the rock reach half its maximum height?
(d) When did the rock hit the surface of the moon again?
Ex. 2 A rock is thrown from the top of an 80-foot building with an initial upward velocity of 64 ft/sec. The height
of the rock is determined by h t   16t 2  64t  80 , where h is measured in feet and t is measured in seconds.
(a) How long did it take the rock to reach its highest point?
(b) How high did the rock go?
(c) When did the rock reach half its maximum height?
(d) When did the rock hit the ground?
2x  1, x  2
A
Graph the function, f x    2
.
x2

x ,
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Use the Max or Min Value of a Quadratic Function to Solve Applied Problems (Vertical Motion)
H
Worksheet 15
#18 Thursday 9/25_______________________________
M
Complete the following statements for each function:
As x  , f x  ____.
As x  , f x  ____.
1. (a) f x   2x 3
(b) f x   2x 3
(c) f x   x 5
(d) f x   0.5x
2. (a) f x   3x 4
(b) f x   0.6x 4
(c) f x   2x 6
(d) f x   0.5x 2
Describe the patterns you observe. In particular, how do the values of the coefficient an and the degree n affect
A
T
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the end behavior of f x   an x n
Use a graphing utility to approximate (round to three decimal places) the local maxima and local minima of
f  x   x3  2 x 2  4 x  5 .
- A polynomial function is a function in the form f  x   an xn  an1 xn1  ...  a 1 x  a0 , where an , an1 ,... a1 , a0
are real number and n is a nonnegative integer. The domain of all polynomials is ALL real numbers.
- End Behavior of Polynomial Functions; The end behavior of a function depends on the degree and leading
coefficient of the polynomial.
None
#19 Friday 9/26 ________________________
M
Ex. 1 Describe the end behavior of f x   x 3  2x 2 11x 12 .
Ex. 2 Describe the end behavior of f x   2x 4  2x 3  22x 2 18x  35 .
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T
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Describe how to transform the graph of f x   x into g x   
1
x 1  2 .
2
Short Quadratics Quiz!
End Behavior of a Function.
Worksheet 16
#20 Monday 9/29_________________________________
M
Ex. 1 Find a polynomial of degree 3 whose zeros are 3, 2, and 5 . Draw a possible graph of this function.
Ex. 2 Is x  1 a factor of f  x   x3  x 2  x  1 ?
4
1

Ex. 3 For the polynomial f x   5 x  2 x  3  x   , state the zeros and their multiplicities. Draw a

2
2
possible graph of f x .
A
T
Ex. 4 For the polynomial f x   x 2 x  2 
a) Find the x- and y-intercepts of the graph of f.
b) Using a graphing calculator, graph the polynomial.
c) For each x-intercept, determine whether it is of odd or even multiplicity
Graph f x   x 2  4x  5 .
Identify the Zeros of a Polynomial Function and Their Multiplicity.
The Factor Theorem: A polynomial function f  x  has a factor x  k if and only if f  k   0 , where k is an x-
intercept of the graph of f x .
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Worksheet 17
#21 Tuesday 9/30_________________________________
M
Ex. 1 Find the zeros (roots) of f  x   x3  x 2  6 x
Ex. 2 Find the zeros (roots) of f  x   x3  36 x
Ex. 3 Find the zeros (roots) of f  x   3x3  x 2  2 x
Ex. 4 Find the zeros (roots) of f  x   x3  3x 2  4 x  12
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Find the vertex and axis of symmetry of the parabola determined by the function f x   3x 2  4x  7 .
Finding the Zeros of a Polynomial Function by Factoring
Finding the real-number zeros of a function f is equivalent to finding x-intercepts of the graph of y  f x  or the
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solutions to the equation f x   0 .
Worksheet 18
#22 Wednesday 10/1 _________________
3
2
M
Ex. 1 Graph f x   x  2 x 1 by finding the x-intercepts, y-intercept and end behavior.
Ex. 2 Graph f x   3x 3  x 2  2x by finding the x-intercepts, y-intercept and end behavior.
Ex. 3 Graph f x   x 4  5x 2  4
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T
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Factor 6x 3  22x2 12x
Graphing Polynomials using x-intercepts (with multiplicity), y-intercept and end behavior.
Worksheet 19
#23 Thursday 10/2_________________________________
M
Review the following topics: Graphing Quadratics; Solve Applied Quadratic Problems; End Behavior of a
Polynomial Function; Zeroes of a Polynomial; Graphing Polynomials
1
A
In the set {2,  2, 0, , 4.5,  } , name the numbers that are integers. Which are rational numbers?
2
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Quadratics and Polynomials Review – Test on Friday
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Worksheet 20
#24 Friday 10/3 ______________________
M
Good luck on Today’s Test
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None
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Test #3 - Quadratics and Polynomials Test
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None
#25 Monday 10/6___________________________________
M
Ex.1 List the possible rational zeros for the function, f x   3x 3  4x 2  5x  2 . Find all rational zeros.
Ex. 2 List the possible rational zeros for the function, f x   2x 3  x 2  9x  9 . Find all rational zeros.
Ex. 3 Find all zeros of f  x   x4  x3  3x2  x  2 .
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H
2x 3  5x 2  6x
2x
Rational Zeros Theorem: Using the constant and the leading coefficient, we can develop a list of all potential
rational (fractional) zeros
Worksheet 21
Rewrite the expression as a polynomial in standard form:
#26 Tuesday 10/7___________________________________
M
Adding and Subtracting Complex Numbers
1) 7  3i  4  5i 
2) 2  i  8  3i 
Multiplying Complex Numbers
4) 2  3i 5  i 
5) 3  5i 4i
Raising a Complex Number to a Power
2
1
3 
6)  
i
2 2 
Complex Conjugates and Division
7) 4  5i 4  5i 
A
8)
5i
2  3i
9) Solve x2  x 1  0
Perform the indicated operation on the complex ( a  bi ) numbers:
3) 8i  4  3i 
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H
1) 2  i 3  2i 
Complex Numbers
Worksheet 22
2) 4  3i 4  3i 
3) 5  i 5  i 
#27 Wednesday 10/8_________________________________
M
Ex. 1 One zero of f x   x 4  5x 3  3x 2  43x  60 is 2  i . Find the other zeros (real and nonreal)
Ex. 2 Write a polynomial function of minimum degree in standard form with real coefficients whose zeros
include x  1 and x  3 i .
Ex. 3 Write a polynomial function of minimum degree in standard form with real coefficients whose zeros and
their multiplicities are x  1 (multiplicity 2) and x  3  i (multiplicity 1)
A
List all potential rational zeros of f x   3x 4  5x 3  3x 2  7x  2
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Fundamental Theorem of Algebra: A polynomial function of degree n has n complex zeros (real and nonreal).
Some of these zeros may be repeated.
Complex Conjugate Zeros
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None
#28 Thursday 10/9___________________________________
M
Ex. 1 The complex number z  1 2i is a zero of f x   4x 4 17x 2 14x  65 . Find the remaining zeros of
f x , and write it in factored form.
Ex. 2 Find all of the zeros and write a linear factorization of f x  if z  1 i is a zero of
f x   x 4  2x 3  x 2  6x  6 .
A
T
H
Ex. 3 Find all zeros of f x   x 5  3x 4  5x 3  5x 2  6x  8 .
A polynomial of degree 5 can have at most ____ distinct real zeros.
Finding Complex Zeros
Worksheet 23
#29 Friday 10/10_____________________________________
M
Ex. 1 Write f x   3x 5  2x 4  6x 3  4x 2  24x 16 as a product of linear and irreducible quadratic factors, each
with real coefficients.
Ex. 2 Write f x   x 3  x 2  x  2 as a product of linear and irreducible quadratic factors with real coefficients.
Ex. 3 Write a polynomial function of minimum degree in standard from with real coefficients whose zeros
include: 1, 2, and 1 i .
A
Find the vertex and axis of symmetry of the graph of f x   x  2 x  6 
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Factoring Polynomial Functions; Review for our next Test (Tuesday, 10/14)
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Worksheet 24