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# Year 13 IB Math Studies - Functions

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## Steve Myers

on 10 September 2013

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#### Transcript of Year 13 IB Math Studies - Functions

Year 13 IB Mathematical Studies
Functions
The Function Machine
To understand functions well, it's helpful to think of it as a machine that takes one kind of thing and turns it into another kind of thing.
Function Mappings
Domain
The set of objects or numbers to be put into a function
Range
The set of objects or numbers that are the output of a function
Different Names for the
Same Thing
Domain
Input
Pre-image
Range
Output
Image
Function Notation
If
then
Everytime we see x, we must replace it with -2 in brackets.
Example 1
Construct a mapping for the function
on the domain
Example 2
Example 3
Example 4
Linear Functions
Linear Function
Any function of the form ,
Example 5
Any function of the form ,
The graph of a quadratic function is a parabola. If a > 0, the graph opens up. If a < 0, the graph opens down.
Vertex
The maximum or minimum point on a parabola. Also called turning point.
Axis of Symmetry
A vertical line that passes through the vertex of a parabola. The equation for the axis of symmetry is
.
.
Intercepts
and
x
y
The y-intercept is simply y = c.
The x-intercepts (also called zeros) can be found in two ways:
Factorization
Using technology
x-intercepts:
y-intercept:
-1 and 3
-2
Example 7
Example 8
Using the axis intercepts and vertex, sketch the graph of
Using a GDC, sketch the graph of
Label the vertex and axis intercepts.
Example 6
a) Find the equation of the line of symmetry of .

b) Find the coordinates of the vertex.
Exponential Functions
Opening Problem
Would you rather be given 20,000 Kc every day of the month, or be given 1 Kc on the first day, 2 Kc the second day, 4 Kc the third day, and have it continue to double every day for the whole month?
Exponential Function
A function in which the variable appears in the exponent
For example, the function representing the doubling of money from the opening problem was:
where is the number of days after the first day.
This is an exponential function because the variable appears in the exponent.
t
Example 1
Asymptotes
Consider the equation .
Make a table of values and graph.
Notice as x gets smaller, y gets really close to 0. This equation has a at y = 0.
horizontal asymptote
Investigation
Complete Investigation: Exponential Graphs on p. 584 in your textbook.
Exponential Growth
Asymptote
An invisible horizontal or vertical line that a graph approaches but never reaches
Example 2
Exponential Decay
Example 3
Big Idea: Graphs of Exponential Functions
From you investigation, you should have figured out that for :
controls the steepness of the graph
controls the steepness of the graph as well
controls the vertical translation and
is the vertical asymptote
If , then the graph is above the asymptote.
If , then the graph is below the asymptote.
Investigation: Understanding Exponential Functions
Filip recently did a photo shoot for the new men's fragrance from Dolce & Gabanna. When his photo was published the D&G's stock was at €36 per share. Each week after the release of the photo, the stock increased by 3.4%.
Problem:
a) Write an exponential function for the value of D&G's stock t weeks after the release of Filip's photo.
b) What was the value (to the nearest cent) of one share of D&G's stock 7 weeks after ...
i) the first 7 weeks?
ii) the first year?
c) How many days (to the nearest day) after the release of the photo did it take the stock rise to €50?
More Example Ideas:
Growth
Cell division
Compound interest
When Mr. Myers leaves his coffee mug over the weekend and mold grows in it
Periodic Funcitons
The London Eye
Imagine you're riding in the London eye and record your height above ground every three seconds. Since you're traveling in a circle, your heights would repeat as you go around. Graphing your height against time would give you a sine wave.
Periodic Data -
data that repeats itself in cycles
Periodic data appears EVERYWHERE:
Weather patterns
Tides
Car engines revolutions
Lunar phases
Animal populations
Hours of daylight
Can you think of anything else?
We can see the repeating pattern easily in the graphs of periodic data:
Period -
the length of one cycle
Amplitude -
the distance from the max or min to the principal axis
Principal axis
Amplitude =
Quick Check: Periodic Basics
The Sine Curve
The period of y = sin x is 360 degrees.
The maximum is 1.
The minimum is -1.
You need to memorize this.
Investigation
Complete Investigations 1 and 2 on page 564 in your book.
You will need a calculator!
Unfamiliar Functions
We will use things we already know about functions and apply it to unfamiliar functions to solve problems. We will use:
axes intercepts
turning points (maxima and minima)
asymptotes
Example
Asymptotes
Horizontal Asymptotes
Horizontal asymptotes occur when a function approaches a value from above or below (ie. exponential functions)
Vertical Asymptotes
Vertical asymptotes occur when you have a function with the variable in the denominator (dividing by zero is undefined)
Horizontal asymptote (graph approaches -1 from above AND below)
Vertical asymptote at x=2 (when x=2, f(2) is 2/(2-2) or 2/0, which is undefined
Example