1.3(TW) - Even and Odd Functions and Symmetry: Bafiniiian...

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Unformatted text preview: Even and Odd Functions and Symmetry: Bafiniiian (I? Ever: anti Martians The Martian f is as. era-n inns-sins if a} W fix] for ail; J; in tits domain at" f. 111;: right sis]: {if the narration {if an even inaction fine-s not change if :r is repiared with Mar. The insisting}. f is an tutti insisting if fl: Mr} V" minim} for aii .r in the domain at? f. tire-r? ram: is the right side of the emanation afar: arid inherits-i changes its sign ii" .31: is replaeeri with. Mr. Eran Fanstians and y-Asis Symmetry The gram]. at? an even i’nnstinn in which ft: if} fix} is. synaraairic with respect. in the pines-is. 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‘s-01C. Wish/M *3 I?) 7’3 a Li 2: X —-X +i Pt) $00: X3,” :3) «F00 a if I awn—nice) rF(——x)= H) " H” J“ :._ _X3___X 1 X2— :2 —(X3+x) 2-. 400 1 "- ifCK) MW- 6937 Viji cewise Func ans 4 W e ace—2): " (/ Functions and Difference Quotients fl efimifiun {if a .ifi‘meme finnfiem The exltrr-eeeimim we 9% m fit in m 33 at 1:]? éwalle-zi 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 X-values. 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 real-life 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)(XS-4). 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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1.3(TW) - Even and Odd Functions and Symmetry: Bafiniiian...

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