symmetry - Math 116 Basic Elementary Functions...

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Unformatted text preview: Math 116 Basic Elementary Functions Transformations of the Functions. Symmetry. CH 1.7 Lectures #5 I Basic elementary functions. Constant function Identlty funCtIOIl I I I I I I I I I i I I t I I I II“V"-'£"‘Q“"l"“v"“l""("‘ ~Iu~uuy~~»~u~I~—y~I I l E I I I I I I I I I l I l I In-n-Iv-Iw-I--s--—q- --I----I--I--I-«xI-ne-I I I I I I I I I I I I I I I I I I~~In~-I-—I~II~~nvII«I~ ~I~—r«I-v~I—-I~~I»~,—I I I I I I I I I I I I I I I I I I“"I"‘1'~'5~'I'--I*"r-1~---I--e-r--I--I--I-"r-I I I I I I I I I I I I I I I I I Basic Quadratic Function I . I I I . I IIII...I..I-.I-III. I I I I I I I I. I I I I I I I I I I. MINIMUMWIIII. I" 2 I. I x :x I I I . I I I I I I I- I 'I" “x'”("‘r"'r""n"':"’f"I I" Math 116 Basic Elementary Functions. Transformations of the Functions. Symmetry. CH 1.7 Lectures #5 I Even and Odd Functions and Symmetry. The function f is an even function if f (~x) = f (x) for all x in the Domain of f. The graph of every even function has a symmetry about y-aXis. That is with any point (x, )2) on the graph, the point (— x, y) is also in the graph, The function f is an odd function if f (—x) = —— f (x) for all x in the Domain of f The graph of every odd function has a symmetry about origin. With any point (x, y) on the graph, the point (— x,— y) is also in the graph. If, for any point (x, y) on the graph, the point (x,— y) is also in the graph, then the graph is said to be symmetric with respect to the x-aXis. Problem#1. Which of basic elementary functions are even? odd? neither even nor odd? Problem#2. Test functions for symmetry. a) f(x)=2x2—x b) g(x)=[3x—4[ c) f(x)=2x2—x4 Problem#3. Is the function f (x) = x3 — 3x2 + x — 10 odd? even? neither odd nor even? Math 116 Basic Elementary Functions. Transformations of the Functions. Symmetry. CH 1.7 Lectures #5 I Functions transformations and graphs. I Vertical Translations. If f is a function and c is a positive constant, then the graph of y : f(x)+ c is the graph of y = f(x) shifted up vertically 0 units, and the graph of y = f (x) ~— 0 is the graph of y = f (x) shifted down vertically (2 units. In coordinates: If the point (x0, yo) in on the graph of f , then the point (x0,y0 i c) is on the graph ofy = f(x)i 0. Problem #4. Given the graph of a function y = f (x)- Sketch the graph of the function y = f (x) — 2. State the domain and range for each function. Math 116 Basic Elementary Functions. Transformations of the Functions. Symmetry. CH 1.7 Lectures #5 Problem #5. Given the graph of a function y = f (x)- Sketch the graph of the function y = f (x) + 2. State the domain and range for each function. I Horizontal Translations. If f is a function and c is a positive constant, then the graph of y = f(x+c) is the graph of y = f(x) shifted left horizontally 0 units, and the graph of y = f (x — c) is the graph of y = f (x) shifted right horizontally 0 units. In coordinates: if the point (xwyo) in on the graph of f , then the point (x0 i 0, yo) is on the graph of y=f(xic). Math 116 Basic Elementary Functions. Transformations of the Functions. Symmetry. CH 1.7 Lectures #5 Problem #6. Given the graph of a function y = f (x)- Sketch the graph of the function y = f (x +1). State the domain and range for each function. Problem #7. Given the graph of a function y = f (x)- Sketch the graph of the function y = f (x — 1). State the domain and range for each function. Math 116 Basic Elementary Functions. Transformations of the Functions. Symmetry. CH 1.7 Lectures #5 Problem #8. Given the graph of a function y = f (x)- Sketch the graph of the function y = f (x +1)—~ 2. State the domain and range for each function. '1' Reflections of Graphs. The graph of ' y = — f (x) is the graph of y = f (x) reflected across the x-axis. ' y = f ( — x) is the graph of y = f (x) reflected across the y-axis. In coordinates: if the point (x0, yo) in on the graph of f then the point (x0,-y0) is on the graph of y = ——f(x), and (—xo,y0) is on the graph of y=f(—x). Math 116 Basic Elementary Functions. Transformations of the Functions. Symmetry. CH 1.7 Lectures #5 Problem #9. Sketch the graphs of y = x2 and y = —-x2. State the domain and range for each function. (Each square is one by one unit.) Problem #10. yzflfindyz—fi. State the domain and range for each function Math 116 Basic Elementary Functions. Transformations of the Functions. Symmetry. CH 1 7 Lectures #5 Problem #11. 1ven egrap 0 a unc 101’] 9 t i . t . l . u . “ J ‘ .Ah Luca...‘ :...‘ 5(60 6 ra o . ., . , y— x , . u . . , . ( A . '...:....'...‘.....:.»:h..n .6 :«.J FWWWM WW , ,mé . , . b __ "~>Iv¥cedr .um»...«~”«~ { HHaMH-Wsz»! . A v ) x » i 1 I x r n I a: - |.«,?gg£,wfly..¥a Umxxix. ....,5. .} .1,y,igg,‘...‘i.,rl Each square IS one by one . unit. v .. u.,. .i...l.. a .-~.-.J. ‘...L..J..V‘~..»..g i.w.x.‘.e.,.‘....a‘s .uuu. .-a..a ~ 9.2. 1 ..‘..._i. ....k....*...i“ z..: 4...:..:....I....': v... “21.: “2" ’1‘ .' m ‘ a 9 i 1 g l . a A 4 t. q a 1 ‘.I».. .x....',.( a .u..:..i. .,.->-..:J~.a.t...ir..a €.a.a..a...n.v.x m...._ .~m4.1 w.’2.. -:.. L-. 1 2 ....‘..——i.. .n.’L.uJ...‘-..Zw.l '...t‘..‘.-A‘.-..$ ’L... ~.Z..vd w -. ! i v‘ x s ’v v . . Maw 1 mm. W m y - a 1 v x s i Hummum,Humans—Hg“ ...K.H;:H~u~.$.n¢wAnny”! (“awunnusnfiuwwh. .s. Hans-Hz vs ..z..:.._’.(JULULJH...’...:..In..a...‘..u. a...‘ tutu-nut..5...X.~.'...:.....'...J..J...z 3 Math 116 Basic Elementary Functions. Transformations of the Functions. Symmetry. CH 1.7 Lectures #5 I Nonrigid transformations. Stretching and Shrinking (vertically and horizontally) graphs. Stretching and shrinking (compression) change the shape of the graph, thus these transformations are called nonrigid. Vertical Stretching/Shrinking I Given a function y = f Will consider the new function y = c f (x), where c is a positive constant, c 721. If 0 <1, the graph of the new function y = c f (x) is a vertically compressed (shrunk) version of the graph of y = f If 0 >1, the graph of the new function y = c f (x) is a vertically stretched version of the graph of y = f In coordinates : if the point (x0, yo) in on the graph of f , then the point (x0,cy0) is on the graph of y = cf(x). Math 116 Basic Elementary Functions. Transformations of the Functions. Symmetry. CH 1.7 Lectures #5 Horizontal Stretching/Shrinking I Given a function y = f Will consider the new function y = f (ex), where c is a positive constant, 0 ¢ 1. If 0 <1, the graph of the new function y = f (c x) is a horizontally compressed (shrunk) version of the graph of y = f(x). If 0 >1, the graph of the new function y = f (ex) is a horizontally stretched version of the graph of y = f (x)- In coordinates : if the point (x0, yo) in on the graph of f, then the point (£9, yo) is on the graph of y = f (0x). 0 10 Math 116 Basic Elementary Functions. Transformations of the Functions. Symmetry. CH 1.7 Lectures #5 Example #6 (p. 159 textbook). The graph of a function y = f (x) is shown below. No formula for f is given. Graph each of the following. a>g<x>=2f<x> b> h<x>=§f<x> f(2x) d) Sp): f[—1-x] 2 O v IN A >< V II 11 Math 116 Basic Elementary Functions. Transformations of the Functions. Symmetry. CH 1.7 Lectures #5 12 Math 116 Basic Elementary Functions. Transformations of the Functions. Symmetry. CH 1.7 Lectures #5 Problem #12. Given the graph of a function f, sketch the graph of 1 a)y=—2-f(x), b)y=3f(x). c) State the domain and range for the original and transformed functions. (Each square is one by one unit.) 13 ...
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