Pre-Cal 40S (Winter '07)
Math Dictionary Notes
Transformations
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Darren
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Kuropatwa
Title: Mar 9-2:47 PM (1 of 25)

Translations y = ƒ(x - a) + b
The role of parameter a: The role of parameter b:
a > 0 the graph shifts RIGHT a units. b > 0 the graph shifts UP b units.
- the x-coordinates are increased a units. - the y-coordinates are increased b units.
a < 0 the graph shifts LEFT a units. b < 0 the graph shifts DOWN b units.
- the x-coordinates are decreased a units. - the y-coordinates are decreased b units.
WARNING: watch the sign of a
Examples:
Title: Mar 9-2:52 PM (2 of 25)

Stretches and Compressions y = aƒ(bx)
The role of parameter a:
|a| > 1 the graph of the function ƒ(x) is stretched vertically.
0 < |a| < 1 the graph of the function ƒ(x) is compressed vertically.
- the y-coordinates are multiplied by a factor a.
Examples:
Title: Mar 11-2:21 PM (3 of 25)

Stretches and Compressions y = aƒ(bx)
The role of parameter b:
|b| > 1 the graph of the function ƒ(x) is compressed horizontally.
0 < |b| < 1 the graph of the function ƒ(x) is stretched horizontally.
- the x-coordinates are multiplied by a factor .
Examples:
Title: Mar 11-3:56 PM (4 of 25)

Reflections
Given any function y = ƒ(x)
-ƒ(x) produces a reflection over the x-axis
The y-coordinates of ƒ(x) are multiplied by -1.
ƒ(-x) produces a reflection over the y-axis.
The x-coordinates of ƒ(x) are multiplied by -1.
Title: Mar 11-6:17 PM (5 of 25)

Putting it all together ...
y = ƒ(x)
Remember: stretches before translations
Title: Mar 9-2:32 PM (6 of 25)

Inverses ...
Algebraically speaking ...
The concept ...
Conceptually analyzing the function ...
Note:
Numerically speaking ...
Graphically speaking ...
Title: Mar 9-2:32 PM (7 of 25)

EVEN FUNCTIONS
Graphically: A function is "even" if its graph is symmetrical about the origin.
These functions
are even...
These are
not ...
Symbolically (Algebraically)
a function is "even" IFF (if and only if) ƒ(-x) = ƒ(x)
Examples: Are these functions even?
1. ƒ(x) = x² 2. g(x) = x² + 2x
ƒ(-x) = (-x)² g(-x) = (-x)² + 2(-x)
ƒ(-x) = x² g(-x) = x² - 2x
since ƒ(-x) = ƒ(x) since g(-x) is not equal to g(x)
ƒ is an even function g is not an even function
Title: Mar 9-2:33 PM (8 of 25)

ODD FUNCTIONS
Graphically: A function is "odd" if its graph is symmetrical about the origin.
These
functions
These are
are odd ... not ...
Symbolically (Algebraically)
a function is "odd" IFF (if and only if) ƒ(-x) = -ƒ(x)
Examples: 1. ƒ(x) = x³ - x 2. g(x) = x³- x²
ƒ(-x) = (-x)³ - (-x) g(-x) = (-x)³ - (-x)²
ƒ(x) = -x³ + x g(x) = -x³ - x²
-ƒ(x) = -(x³ - x) -g(x) = -(x³-x²)
-ƒ(x) = -x³ + x -g(x) = -x³+ x²
since ƒ(-x)= -ƒ(x) since g(-x) is not equal to -g(x)
ƒ is an odd function g is not an odd function
Title: Mar 9-2:34 PM (9 of 25)

Sketching reciprocal function graphs
IMPORTANT:
i.e a reciprocal is not the
#3
same thing as an inverse
#1
#3
Invariant
Points #2
#3
#1
#3
Title: Mar 9-2:36 PM (10 of 25)

Example:
Sketch the graph
Title: Mar 9-2:37 PM (11 of 25)

The Reciprocal Trigonometric Functions ...
ƒ(x) = csc(x) ƒ(x)= 1/sin(x)
Title: Mar 9-2:37 PM (12 of 25)

The Reciprocal Trigonometric Functions ...
ƒ(x)= sec(x) ƒ(x)= 1/cos(x)
Title: Mar 9-2:37 PM (13 of 25)

The Reciprocal Trigonometric Functions ...
ƒ(x)= cot(x) ƒ(x)= 1/tan(x)
Title: Mar 9-2:37 PM (14 of 25)

Sketching the graphs of absolute value functions ...
Step 1: Sketch the graph of y = ƒ(x).
Step 2: Reflect all the part of the line that
y=x
is below the x-axis over the x-axis.
Title: Mar 9-2:38 PM (15 of 25)

Trigonometric Modeling
and Transformations
An Example
For a Saskatchewan town
the latest sunrise is on Dec
21 at 9:15 am. The earliest
sunrise is on June 21 at 3:15
am. Sunrise times on other
dates can be predicted using
a sinusoidal equation.
Note: There is no daylight
savings time in Morning at Swiftcurrent Lake
photo source: http://www.flickr.com/photos/58518845@N00/381683114
Saskatchewan.
a) Sketch the graph of the sinusoidal function described above.
b) Write 2 equations for the function; one using sine the other cosine.
c) Use one of the equations in (b) to predict the time of sunrise on April 6.
d) What is the average sunrise time throughout the year?
e) On what days will the sunrise at 7:00am?
Title: Mar 9-2:39 PM (16 of 25)

Trigonometric Modeling and Transformations
An Example
For a Saskatchewan town the latest sunrise is on Dec 21 at 9:15am. The earliest
sunrise is on June 21 at 3:15am Sunrise times on other dates can be predicted
using a sinusoidal equation.
Note: There is no daylight savings time in Saskatchewan.
a) Sketch the graph of the sinusoidal function described above.
Title: Mar 9-2:40 PM (17 of 25)

Trigonometric Modeling and Transformations
An Example
For a Saskatchewan town the latest sunrise is on Dec 21 at 9:15am. The earliest
sunrise is on June 21 at 3:15am Sunrise times on other dates can be predicted
using a sinusoidal equation.
Note: There is no daylight savings time in Saskatchewan.
b) Write 2 equations for the function; one using sine the other cosine.
Title: Mar 9-2:40 PM (18 of 25)

Trigonometric Modeling and Transformations
An Example
For a Saskatchewan town the latest sunrise is on Dec 21 at 9:15am. The earliest
sunrise is on June 21 at 3:15am Sunrise times on other dates can be predicted
using a sinusoidal equation.
Note: There is no daylight savings time in Saskatchewan.
c) Use one of the equations in (b) to predict the time of sunrise on April 6.
Title: Mar 9-2:41 PM (19 of 25)

Title: Mar 9-2:41 PM (20 of 25)

Trigonometric Modeling and Transformations
An Example
For a Saskatchewan town the latest sunrise is on Dec 21 at 9:15am. The earliest
sunrise is on June 21 at 3:15am Sunrise times on other dates can be predicted
using a sinusoidal equation.
Note: There is no daylight savings time in Saskatchewan.
d) What is the average sunrise time throughout the year?
Title: Mar 9-2:41 PM (21 of 25)

Trigonometric Modeling and Transformations
An Example
For a Saskatchewan town the latest sunrise is on Dec 21 at 9:15am. The earliest
sunrise is on June 21 at 3:15am Sunrise times on other dates can be predicted
using a sinusoidal equation.
Note: There is no daylight savings time in Saskatchewan.
e) On what days will the sunrise at 7:00am?
Title: Mar 9-2:42 PM (22 of 25)

Title: Mar 9-2:42 PM (23 of 25)

Title: Mar 9-2:42 PM (24 of 25)

How many days of the year will the sun rise later than 7 am?
Title: Mar 9-2:43 PM (25 of 25)