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R Kennedy 1
Chosen as the EXCLUSIVE publisher for the newPre-Calculus Grade 11& 12 courses
Opening Doors!
Calgary Teacher’sConventionFebruary 16, 2012
Table of Contents
• Unit 1: Transformations and Functions– Chapter 1: Function Transformations– Chapter 2: Radical Functions– Chapter 3: Polynomial Functions
• Unit 2: Trigonometry– Chapter 4: Trigonometry and the Unit Circle– Chapter 5: Trigonometric Functions and Graphs– Chapter 6: Trigonometric Identities
• Unit 3: Exponential and Logarithm Functions– Chapter 7: Exponential Functions– Chapter 8: Logarithmic Functions
• Unit 4: Equations and Functions– Chapter 9: Rational Functions– Chapter 10: Function Operations– Chapter 11: Permutations, Combinations, and the Binomial Theorem
• Unit• Unit Opener• Unit Project
• Chapters (2 or 3 per unit)• Sections (3 to 5 per chapter)• Investigate• Link the Ideas• Check Your Understanding• Chapter Review • Practice Test
• Unit Project Wrap-Up• Cumulative Review• Unit Test
3-Part Lesson
Math 30-1 Possible Course Outline September 2012 – January 2013
Chapter Number of Days Tentative Exam Date
Function Transformations 8 September 13
Radical Functions 6 September 21
Polynomial Functions 8 October 3
Trig and the Unit Circle 8 October 16
Trig Functions and Graphs 8 October 26
Trig Identities 8 November 6
Exponential Functions 6 November 16
Log Functions 7 November 27
Rational Functions 6 December 5
Function Operations 6 December 13
Perms/Combs 6 December 21
R Kennedy 8Pre-Calculus 12, McGraw-Hill Ryerson
Chapter 1 Transformations
1.1
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A variable y is said to be a function of a variable x if there is a relation between x and y such that every value of x corresponds to one and only one value of y.
What is a Function?
The symbol ‘ f (x) ’ may be used to denote a function of x.
For example, 4x + 5 is the function of x. It can be expressed as f(x) = 4x + 5.
22 )(,)( xxgxxh or 2)( xxF
Besides x, we can have functions of other variables, forexample,
543 2 uu is a function of u and we may write
.543)( 2 uuuf
The letter ‘f ’ in the symbol ‘f(x)’ can be replaced by other letters,for example,
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Functions
linear
quadratic absolute value square root
cube root
cubic
reciprocal exponential
logarithmic
sine
1.1.2
Graphs of Functions
cosineLine Dance
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y = |x|+ 8
y = |x| – 8
Graph Translations of the Form y – k = f(x)Given the graph of y = |x|, graph the functions y = |x| + 8 and y = |x| – 8.
The transformed graphs are congruent to the graph of y = |x|. Each point (x, y) on the graph of y = |x| is transformed to become the point (x, y + 8) on the graph of y = |x| + 8.Ex: (–4, 4) (–4, 12)
It becomes the point (x, y – 8) on the graph of y = |x| – 8.Ex: (–4, 4) (–4, -4)
(-4, 4)
(-4, 12)
(-4, -4)
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Graphing y = f(x) + k
x
–8
–6
–4
–2
0
2
4
6
8
y = |x|
8
6
4
2
0
2
4
6
8
y = |x|+8
16
14
12
10
8
10
12
14
16
The graph of a function is translated vertically if a constant is either added or subtracted from the original function. Each point (x, y) on the graph of y = |x| is transformed to become the point (x, y + 8) on the graph of y – 8 = |x|. Using mapping notation, (x, y) → (x, y + 8).
Graph y = |x| + 8
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y = |x + 7| y = |x - 8|
Graph Translations of the Form y = f(x – h)Given the graph of y = |x|, graph the functions y = |x + 7| and y = |x – 8|.
The transformed graphs are congruent to the graph of y = |x|. Each point (x, y) on the graph of y = |x| is transformed to become the point (x – 7, y) on the graph of y = |x + 7|.Ex: (–4, 4) (–11, 4)
It becomes the point (x + 8, y) on the graph of y = |x – 8|.Ex: (–4, 4) (4, 4)
(-4, 4)(-11, 4) (4, 4)
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Horizontal and Vertical TranslationsSketch the graph of y = |x – 4|– 3.
• Apply the horizontal translation of 4 units to the right to obtain the graph of y = |x – 4|.
• Apply the vertical translation of 3 unitsdown to y = |x – 4| to obtain the graphof y = |x – 4| – 3.
y = |x – 4| – 3
y = |x – 4|
The point (0, 0) on the function y = |x| is transformed to becomethe point (4, -3). In general, the transformation can be described as(x, y) → (x + 4, y – 3).
(0, 0)
(4, -3)
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• Given the functiondefined by a table
• Determine the coordinates of the following transformations
x –3 –2 –1 0 1 2 3
f(x) 7 4 9 3 12 5 6
f(x) + 3
f(x + 1)
f(x – 2) + 4
Transformation of functions y – k = f(x – h)
(0, 6)(–3, 10) (–2, 7) (–1, 12) (1, 15) (2, 8) (3, 9)
(–1, 3)(–4, 7) (–3, 4) (–2, 9) (–2, 12) (–3, 5) (–4, 6)
(2, 7)(–1, 11) (0, 8) (1, 13) (3, 16) (4, 9) (5, 10)
Each point (x, y) on the graph of y = f(x)is transformed to become the point (x + h, y + k) on the graph of y – k = f(x – h). Using mapping notation, (x, y) → (x + h, y + k).
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Transformation of functions y – k = f(x – h)
Possible Assignment
Essential: #1 – 3, 5, 6, 8, 10 – 12, C1, C2, C4Typical: #5, 7 – 12, 13 or 14, C1, C2, C4Enrichment #15 – 19, C2 – C4
What might emerge if we let kid loose to work on anything they want for a day with the only proviso that their presentation the next day explain:why they had undertaken this work,how it used or connected with math andwhat they had done?
There is an Australian software company called Atlassian and they do something once a quarter where they say to their software developers: You can work on anything you want, any way you want, with whomever you want, you just have to show the results to the rest of the company at the end of 24 hours. They call these things Fed-Ex Days, because they basically have to deliver something overnight. That one day of intense autonomy has produced a whole array of software fixes, a whole array of ideas for new products, and a whole array of upgrades for existing products
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