Transcript Pre-Calculus Chapter 6

Chapter 1
Functions and Their Graphs
1.3.2 Even and Odd Functions
 Objectives:
Identify and graph step functions and other
piecewise-defined functions.
Identify even and odd functions.
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Vocabulary
 Step Function
 Greatest Integer Function
 Even and Odd Functions
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Warm Up 1.3.2

During a seven-year period, the population P (in
thousands) of North Dakota increased and then
decreased according to the model
P = –0.76t2 + 9.9t + 618, 5 ≤ t ≤ 11,
where t represents the year, with t = 5 corresponding
to 1995.
a. Graph the model over the appropriate domain using
your graphing calculator.
b. Use this graph to determine which years the
population was increasing. During which years was
the population decreasing?
c. Approximate the maximum population between 1995
and 2001.
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Greatest Integer Function
 Defined as the greatest integer less than or equal to x.
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More Greatest Integer
 The greatest integer function is an example of a step
function.
 Find the following values:
a.
 1 
 1 
b.  
 10 
c.
1 . 5 
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Graphing a Piecewise Function
 Sketch the graph of f (x)
by hand.
 2 x  3, x  1
f ( x)  
  x  4, x  1
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Even and Odd Functions
 Even Function
Symmetric with respect to the y-axis.
For every (x, y) on the graph, there is also (–x, y).
 Odd Function
Symmetric with respect to the origin.
For every (x, y) on the graph, there is also (x, –y).
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Graphs of Function Symmetry
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How Do We Know If It’s
Even, Odd, or Neither?
 Even Function
Graphical: Symmetric about y-axis (mirror image).
Algebraic: For each x in domain of f, f (–x) = f (x).
 Odd Function
Graphical: Image is the same when rotated 180°.
Algebraic: For each x in domain of f, f (–x) = –f (x).
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Example 6
Determine algebraically and graphically whether
each function is even, odd, or neither.
a.
g(x) = x3 – x
b.
h(x) = x2 + 1
c.
f (x) = x3 – 1
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Homework 1.3.2
 Worksheet 1.3.2
# 41, 45, 47, 53 – 71 odd, 79, 80
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