Translations of Data and The Graph
Scale-Change Theorem
Sections 3.3 - 3.4

x1 = 36

Mean: 39.2
Median: 36
Mode: 36
Range: 24
IQR: 14
Variance: 81.198
Standard deviation: 9.011






x2=30 x3=40
x4= 54

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x5=36
Mean: 64.2
Median: 61
Mode: 61
Range: 24
IQR: 14
Variance: 81.198
Standard deviation: 9.011

When there is a translation, the mean, median, and mode
will be affected by the amount of the translation.

The range, interquartile range, variance, and standard
deviation remain unchanged with a translation; they are
invariant under translation.
Symmetries of Graphs!

Reflection-symmetric - Line

180˚ Rotation-Symmetric or PointSymmetric – if it can be mapped to itself under
a rotation of 180˚ around P

Center of Symmetry: Point P.

Power Function: f(x) = xn where n an integer ≥
2
Examples!
Symmetric to the x axis!
Symmetric to the Y-axis!
(x,y) then (-x,y) is also on
The graph!
Symmetric to the Origin!
(x,y) then (-x,-y) is
Also on the graph!
Even functions  symmetric to the y-axis!
A function f is an even function if
f ( x)  f ( x)
for all values of x in the domain of f.
Example:
f ( x)  3x 2
is even because
f ( x)  3( x) 2  3x 2  f ( x)
Odd functions – symmetric to
the origin!
A function f is an odd function if
f ( x)   f ( x)
for all values of x in the domain of f.
Example:
f ( x)  5 x 3  x
is odd because
f ( x)  5( x)3  x  5x3  x  (5x3  x)   f ( x)
Examples!
Prove that the graph of y = x + x3 + 2x5 is
symmetric to the origin.
 To prove, I must show that (a,b) then (-a, -b)
 If b = a + a3 + 2a5 , I must show that –b = -a + (-a3)
+ 2(-a)5

Multiply everything in the first by -1
 -b = -1 (a + a3 + 2a5)
 -b = -1a + -a3 – 2a5
 -b = -a +(-a)3 +2 (-a)5

Example!
 Use
an automatic grapher to conjecture
whether the function is odd, even, or
neither. Then prove your conjecture.
f(x) = 2x3
Looks odd! f(-x) = 2(-x)3
f(-x) = -2x3
Example


Give equations for the asymptotes for
so, x = -2 and y = 4
Homework
Pages 183 – 184
1, 3-5, 8-13