Transcript pps

Learning Objectives for Section 2.2
Elementary Functions;
Graphs and Transformations
 You will become familiar with some elementary functions.
 You will be able to transform functions using vertical and
horizontal shifts.
 You will be able to transform functions using stretches and
shrinks.
 You will be able to graph piecewise-defined functions.
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Six Graphs of Common Functions
___________ Function
___________ Function
Domain: _________
Range: _________
Domain: _________
Range: _________
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Six Graphs of Common Functions (continued)
f ( x)  x
f (x)  x
•
______________ Function
_________________ Function
Domain: _________
Domain: _________
Range: _________
Range: _________
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Six Graphs of Common Functions (continued)
f ( x)  x2
f ( x)  x3
_______________ Function
____________ Function
Domain: _________
Domain: _________
Range: _________
Range: _________
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Vertical and Horizontal Translations
TRANSLATIONS, also called shifts, are simple transformations
of the graph of a function whereby each point of the graph is
translated (shifted) a certain number of units vertically and/or
horizontally.
The shape of the graph remains the same.
Vertical translations are shifts upward or downward.
Horizontal translations are shifts to the right or to the left.
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Vertical and Horizontal Translations
Example
Graph the function on your calculator and describe the transformation
that the graph of f ( x)  x must undergo to obtain the graph of h(x).
h( x)  x  5
h( x)  x  5
h( x)  x  5
h( x)  x  5
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Vertical and Horizontal Translations
Vertical and Horizontal Translations of the Graph of y = f(x)
For h, k positive real numbers:
1. Vertical shift k units UPWARD:
h( x)  f ( x)  k
2.
h( x)  f ( x)  k
Vertical shift k units DOWNWARD:
3. Horizontal shift h units to the RIGHT:
h( x)  f ( x  h)
4.
h( x)  f ( x  h)
Horizontal shift h units to the LEFT:
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Transformations of f(x) = x2
Given
y1  x2

y2  x2  4

y3  x2  4

y4   x  4

y5   x  4
Describe each transformation of the graph of y1.
2
2
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Vertical and Horizontal Translations (continued)
Take a look*
Name the transformations that f(x) = x2 must undergo to obtain the
graph of g(x) = (x + 3)2 – 5
Solution:
You can obtain the graph of g by translating f 3 units ___________
and 5 units ____________________.
f(x)
g(x)
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Reflecting Graphs
A REFLECTION is a mirror image of the graph in a certain line.
Reflection in the x-axis: h(x) = - f(x)
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Reflecting Graphs (continued)
Take a look*
Sketch the graph of f(x)= x and g(x)= - x .
Solution:
You can obtain the graph of g by reflecting f in the x-axis.
f(x)
f(x) = x
g(x)
Parent Function
f(x)= x
Reflection in x-axis
g(x)= - x
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Example
Describe the transformations of each of the following graphs as
compared to the graph of its parent function. Then sketch the graph
of the transformed function.
a)
y  x4
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b)
y   x3  1
c)
y  x 3 2
Nonrigid Transformations
Translations and reflections are called rigid transformations
because the basic shape of the graph is NOT changed.
Nonrigid transformations cause the original shape of the graph
to change or become distorted.
We will look briefly at VERTICAL STRETCHES AND SHRINKS.
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Vertical Stretch and Shrink
A VERTICAL STRETCH causes the graph to become more elongated (skinnier)
A VERTICAL SHRINK causes the graph to become squattier (wider).
For y = f(x),
y
= A f(x) is a vertical stretch if A > 1.
y
= A f(x) is a vertical shrink if 0 < A < 1.
Example:
f(x) = x 2
g(x) = 4x
h(x) =
2
1 2
x
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Notice that the vertex of the parabola
does not change; the graph just
becomes narrower or wider
depending upon the value of c.
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Examples
Describe the transformations of each of the following graphs as
compared to the graph of its parent function. Then sketch the graph
of the transformed function.
a)
y
1
x4
3
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b)
c)
y  5x3  1
y  2 x 3
Summary of
Graph Transformations
 Vertical Translation: y = f (x) + k
 k > 0 Shift graph of y = f (x) up k units.
 k < 0 Shift graph of y = f (x) down |k| units.
 Horizontal Translation: y = f (x+h)
 h > 0 Shift graph of y = f (x) left h units.
 h < 0 Shift graph of y = f (x) right |h| units.
 Reflection in x-axis: y = -f (x)
.
 Vertical Stretch and Shrink: y = Af (x)
 A > 1:
Stretch graph of y = f (x) vertically by multiplying
each ordinate (y-value) value by A.
 0<A<1: Shrink graph of y = f (x) vertically by multiplying
each ordinate (y-value) value by A.
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More Practice
More practice with domain, range, and transformations of
common functions.
See Handout
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Piecewise-Defined Functions
 The absolute value of a real number x can be defined as
 x if x  0
| x|
  x if x  0
 Notice that this function is defined by different rules for
different parts of its domain. Functions whose definitions
involve more than one rule are called piecewise-defined
functions.
 Graphing one of these functions involves graphing each rule
over the appropriate portion of the domain.
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Example of a
Piecewise-Defined Function
Graph the function
x
y
2  2x if x  2
f ( x)  
 x  2 if x  2
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