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Unformatted text preview: Even and Odd Functions and Symmetry: Bafiniiian (I? Ever: anti Martians
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ii" .31: is replaeeri with. Mr. Eran Fanstians and yAsis Symmetry The gram]. at? an even i’nnstinn in which ft: if} fix} is. synaraairic with
respect. in the pinesis. A graph is symmetric with respect to the y—axis if, for every point (X,y) on the graph, the point (x,y) is also on the graph. All even functions have graphs with this kind of symmetry. A graph is symmetric with respect to the origin if, for every point (X,y) on the graph, the point (X,y) is also on the graph. Observe that the first and third
quadrant portions of odd functions are reflections of
one another with respect to the origin. Notice that f(X)
and f(x) have opposite signs, so that f(~x)=f(x). All
odd functions have graphs with origin symmetry. Determine Whether each ra h iven is an even function
an odd function or a ﬁmction that is neither even nor odd Note: Compare these tests to the symmetry tests for equations. B‘s01C. Wish/M *3 I?) 7’3 a Li
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W e ace—2): " (/ Functions and Difference Quotients ﬂ eﬁmiﬁun {if a .iﬁ‘meme ﬁnnﬁem
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m 33 at 1:]? éwallezi dii’fewlwe qwtiwt W4 _ ﬁx} m Minx] Some piecewise functions are called step functions
because their graphs form discontinuous steps. One such
function is called the greatest integer function, symbolized
by int(X) or [X], where int(X)= the greatest integer that is less than or equal to X.
For example, int(l)=l, int(l.3)=l, int(l.5)=l, int(l.9)=l int(2)=2, int(2.3)=2, int(2.5)=2, int(2.9)=2 Graphing: 1. Press the blue “y =” button. Enter the ﬁmction you want to graph.
For example, f(x) = 3x2 — 4X + 1 can be entered as y1 = 3xA2 — 4x + 1.
The symbol “x” can be entered ﬁom the button to the right of the green “alpha” button. If you wish to graph a second function at the same time
as the ﬁrst, you can enter it on the “y2 =” line of the “ =” screen. 2. Now press the blue “graph” button. You should see the graph of a
parabola. To adjust viewing window, press the blue “window” button.
Try changing the xmin, xmax, ymin, and ymax values. Changing these
numbers will change the left, right, bottom, and top bounds of the
viewing window. You must choose xmin < xmax and ymin < ymax.
Press the “graph” button the regraph the function. Making a Table of Function Values: 1. Press the blue “y =” button. Enter the ﬁmction for which you want a
table. Use the same notation as in part 1 of the graphing instructions. 2. Now press the yellow “2nd” button and then the blue “window”
button. This is where you setup the properties of the table. Set TblStart
to the ﬁrst x value you want to see, say 0. The value on the next line
(ATbl — read “delta Table”) represents the increments of the Xvalues. Enter a nonzero number for ATbl. Try a positive value. Later you can
try a negative value.  3. Now press the yellow “2nd” button and then the blue “graph” button.
You will now see a table if input and output values for the function.
Each line (ordered pair) represents a point on the graph of the ﬁinction. Note 4: There are many reallife situations that are not modeled by
functions. For example, systems that exhibit “hysteresis” cannot be
modeled by the functions we encounter in this course. Piecewise—deﬁned functions can be graphed on the TI calculators. On the TI Calculators, graphing these functions uses logical tests (yes = 1,
no = 0). 1. Press the MODE button (next to 2nd) and select DOT mode. This prevents the calculator from connecting parts of a graph that should not
be connected. 2. Press the Y: button. On line y1 = we will enter a function to graph. On line yl, enter y1= (1+X)(3SX)(X<0)+(X2)(OSX)(XS4). Now press
GRAPH. You should see part of a line segment followed by part of a parabola.
The inequality symbols are obtained in the TEST menu (press 2‘"I Math).
The parts in parentheses involving inequality symbols are logical
variables. They are either true or false. If a condition is true, the part in
parentheses is assigned the value 1. If it is false, it is assigned the value
0. In this way, only one part of the function is graphed at one time. One
part is “zeroed” while the other part is graphed. If you are going to major in computer science or do any computer programming, you should
be familiar with logical variables ...
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 Fall '11
 BlisinHestiyas

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