Today in Precalculus
•Need a calculator
•Go over homework
•Notes: Rigid Graphical Transformations
•Homework
Homework answers
2) G
4) A
6) F
8) H
10) C
12) B
20) y = x, x , x ,e ,ln x
x
3
24) y = x, x ,ln x
3
26) y = x, x
3
Rigid Transformations
• Leave the size and shape of the graph
unchanged.
• Types:
– vertical shifts
– horizontal shifts
– reflections
Vertical Translations
• Shift the graph up and down
• Equation is changed by simply adding or
subtracting a constant.
y = f(x) + c or y = f(x) – c
y
y
y
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

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x
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
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

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y=x2


x
x
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
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y=x2+3

y=x2-2







Horizontal Translations
• Shift the graph left and right
• Equation is changed by adding or
subtracting a constant from x. (moves in
opposite direction) y = f(x+c) or y = f(x-c)
y
y
y
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

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x
x






y=x2
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x

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
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



y=(x-2)2

y=(x+3)2




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
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
Practice
• A cosine graph with a vertical shift down 5
• An absolute value graph with a horizontal
shift left 2
• A parabola with a vertical shift up 4 and a
horizontal shift left 3
y = (x+3)2 + 4
y = cos(x) – 5

y
y



y = |x+2|

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



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


x

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
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


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
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


x




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




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


Reflections
• Two graphs that are symmetric with
respect to a line (such as the x- or y-axis)
• To reflect across the x-axis negate the
entire function
y = –f(x)
• To reflect across the y-axis, negate every
x within the function
y = f(–x)
Example
3x  1
Given f ( x)  2
write the equations that will reflect
x 2
the graph across the x  axis, and then the y  axis.
Then graph them to check your answers.

3( x)  1
f ( x) 
( x) 2  2

y
3x  1
f ( x)  2
x 2

x








3x  1
 f ( x)   2
x 2

Misc.
• What happens when an even function is
reflected across the y-axis?
– Identical graphs
• What happens when an odd function is
reflected across both the x- and y-axis?
– Reflections are identical
y
y
 y
 y
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x
x
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x
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x
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Homework
• Pg 147: 2-10 even, 17-19 all, 25, 27, 29, 30
• Memorize 10 basic functions