Lecture 5 - Increasing & Decreasing Functions; Transformation of Functions; Quadratic Functions (2.3 – 2.5) Ex 1 Find domain of f(x) = √(6 – x2) 6 – x2 ≥ 0 -x2 ≥ -6 x2 ≤ 6 x ≤ √6 ; x ≥ -√6 [-√6, √6] Find range of f(x) = [0, √6] Ex. 2 Is the following graph a function? EMBED MSGraph.Chart.8 \s No, it is not a function. Vertical line test – If the graph intersects along a vertical line at more than one point, it is not the graph of a function As x increases or you move right, what happens to y? f is an increasing function if x1 < x2 → f(x1) < f(x2) f is a decreasing function if x1 < x2 → f(x1) > f(x2) Types of functions 1. f(x) = mx + b; linear function HYPERLINK "http://rds.yahoo.com/_ylt=A0WTefcBIddI71EBKviJzbkF;_ylu=X3oDMTBqcmtvYmMyBHBvcwM4NQRzZWMDc3IEdnRpZAM-/SIG=1o65ao24l/EXP=1222144641/**http%3A/images.search.yahoo.com/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dgraph%252Bof%252Blinear%252Bfunction%2526js%253D1%2526ni%253D18%2526ei%253Dutf-8%2526fr%253Dfptb-acer%2526xargs%253D0%2526pstart%253D1%2526b%253D73%26w=345%26h=215%26imgurl=artis.inrialpes.fr%252FMembres%252FXavier.Decoret%252Fresources%252Fscipres%252Fwiki%252Fimages%252F4%252F42%252FLinear-curve.png%26rurl=http%253A%252F%252Fartis.inrialpes.fr%252FMembres%252FXavier.Decoret%252Fresources%252Fscipres%252Fwiki%252Findex.php%252FDirector%2527s_commands%26size=2.7kB%26name=Linear-curve.png%26p=graph%2Bof%2Blinear%2Bfunction%26type=png%26oid=b9d4e5411571e9bc%26no=85%26tt=290%26sigr=135n1bm4v%26sigi=12tov2d6g%26sigb=13unjm5es" INCLUDEPICTURE "http://re3.yt-thm-a03.yimg.com/image/25/m1/1719890326" \* MERGEFORMATINET 2. f(x) = b; constant function HYPERLINK "http://rds.yahoo.com/_ylt=A0WTefRlIddIZPQAz4.JzbkF;_ylu=X3oDMTBpaWhqZmNtBHBvcwMzBHNlYwNzcgR2dGlkAw--/SIG=1k5it5m8i/EXP=1222144741/**http%3A/images.search.yahoo.com/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dgraph%252Bof%252Bconstant%252Bfunction%2526fr%253Dfptb-acer%2526ei%253Dutf-8%2526js%253D1%2526x%253Dwrt%26w=343%26h=298%26imgurl=jwilson.coe.uga.edu%252FEMAT6680%252FHorst%252Fderivativeconstant%252Fimage81.gif%26rurl=http%253A%252F%252Fjwilson.coe.uga.edu%252FEMAT6680%252FHorst%252Fderivativeconstant%252Fderivativeconstant.html%26size=4.5kB%26name=image81.gif%26p=graph%2Bof%2Bconstant%2Bfunction%26type=gif%26oid=548dd449a0360a8a%26no=3%26tt=173%26sigr=12kvt5qnn%26sigi=12105hgf2%26sigb=13aro2lgi" INCLUDEPICTURE "http://re3.yt-thm-a02.yimg.com/image/25/m2/2310144943" \* MERGEFORMATINET 3. f(x) = ax2 + bx + c; quadratic function HYPERLINK "http://rds.yahoo.com/_ylt=A0WTefSvIddIZPQAILGJzbkF;_ylu=X3oDMTBpc2VvdmQ2BHBvcwM3BHNlYwNzcgR2dGlkAw--/SIG=1jj13h66h/EXP=1222144815/**http%3A/images.search.yahoo.com/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dgraph%252Bof%252Bquadratic%252Bfunction%2526fr%253Dfptb-acer%2526ei%253Dutf-8%2526js%253D1%2526x%253Dwrt%26w=422%26h=343%26imgurl=jwilson.coe.uga.edu%252FEMAT6680Fa05%252FTrandel%252Ffa05asgn2%252Fgraph1.jpg%26rurl=http%253A%252F%252Fjwilson.coe.uga.edu%252FEMAT6680Fa05%252FTrandel%252Ffa05asgn2%252Ftext2.html%26size=21.8kB%26name=graph1.jpg%26p=graph%2Bof%2Bquadratic%2Bfunction%26type=JPG%26oid=6d20aaf834b40006%26no=7%26tt=118%26sigr=124glpvm3%26sigi=11tiesgeg%26sigb=13b1tabvq" INCLUDEPICTURE "http://re3.yt-thm-a04.yimg.com/image/25/m3/2599000648" \* MERGEFORMATINET 4. f(x) = x3 INCLUDEPICTURE "http://library.thinkquest.org/2647/media/oddxxx.gif" \* MERGEFORMATINET 5. f(x) = √x HYPERLINK "http://rds.yahoo.com/_ylt=A0WTefg9ItdIj.AAX7aJzbkF;_ylu=X3oDMTBpdnJhMHUzBHBvcwMxBHNlYwNzcgR2dGlkAw--/SIG=1i1j2spr3/EXP=1222144957/**http%3A/images.search.yahoo.com/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dgraph%252Bof%252Bsquare%252Broot%2526fr%253Dfptb-acer%2526ei%253Dutf-8%2526js%253D1%2526x%253Dwrt%26w=284%26h=298%26imgurl=www.math10.com%252Falgimages%252FrootSquareRootGraph.gif%26rurl=http%253A%252F%252Fwww.math10.com%252Fen%252Falgebra%252Fradical.html%26size=3.1kB%26name=rootSquareRootGraph.gif%26p=graph%2Bof%2Bsquare%2Broot%26type=gif%26oid=d54eaaa37b2ce94a%26no=1%26tt=127%26sigr=11dvhne96%26sigi=11gf8ls2t%26sigb=1345r2069" INCLUDEPICTURE "http://re3.yt-thm-a01.yimg.com/image/25/m6/3364230431" \* MERGEFORMATINET 6. f(x) = |x| HYPERLINK "http://rds.yahoo.com/_ylt=A0WTefcBIddI71EBIfiJzbkF;_ylu=X3oDMTBqNmJ0Zzk0BHBvcwM3NgRzZWMDc3IEdnRpZAM-/SIG=1krrfm377/EXP=1222144641/**http%3A/images.search.yahoo.com/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dgraph%252Bof%252Blinear%252Bfunction%2526js%253D1%2526ni%253D18%2526ei%253Dutf-8%2526fr%253Dfptb-acer%2526xargs%253D0%2526pstart%253D1%2526b%253D73%26w=300%26h=300%26imgurl=hotmath.com%252Fimages%252Fgt%252Flessons%252Fgenericalg1%252Fabs_value_graph.gif%26rurl=http%253A%252F%252Fhotmath.com%252Fhelp%252Fgt%252Fgenericalg1%252Fsection_4_4.html%26size=2.5kB%26name=abs_value_graph.gif%26p=graph%2Bof%2Blinear%2Bfunction%26type=gif%26oid=9b419fac70c257c0%26no=76%26tt=290%26sigr=11nfdp1ck%26sigi=11tphu0de%26sigb=13unjm5es" INCLUDEPICTURE "http://re3.yt-thm-a02.yimg.com/image/25/m4/2726016338" \* MERGEFORMATINET Ex. 3 On the graph, when is f increasing? For what values of x? EMBED MSGraph.Chart.8 \s Increasing: (-∞, -2) U (1, ∞) Ex. 4 f(x) = x2; g(x) = f(x) + 2 x f g -3 9 11 -2 4 6 -1 1 3 0 0 2 1 1 3 2 4 6 3 9 11 Ex. 5 Piecewise defined functions f(x) = {2x + 1 if x ≤ 1 { x2 if x > 1 f(3) = 32 = 9 f(5) = 52 = 25 f(-3) = 2(-3) + 1 = -5 f(1) = 2(1) + 1 = 3 Ex. 6 f(x) = |x| = {x if x ≥ 0 {-x if x < 0 The graph of y = f(x) + c has the same shape as y = f(x), only shifted c units up if c > 0 or down if c < 0. (Vertical shift) Ex. 7 f(x) = x2 – 4 EMBED MSGraph.Chart.8 \s Ex. 8 f(x) = x2 g(x) = (x – 2)2 EMBED MSGraph.Chart.8 \s x f g -3 9 25 -2 4 16 -1 1 9 0 0 4 1 1 1 2 4 0 3 9 1 The graph of y = f(x + c) has the same shape as y = f(x), only shifted c units left if c > 0 or right if c < 0 (Horizontal shift) Ex. 9 f(x) = |x – 2| + 4 EMBED MSGraph.Chart.8 \s Ex. 10 y = f(x) = x2 y = -f(x) = -x2 EMBED MSGraph.Chart.8 \s EMBED MSGraph.Chart.8 \s The graph of y = -f(x) is gotten by reflecting the graph of y = f(x) about the x-axis Ex. 11 y = f(x) = √x y = f(-x) = √-x EMBED MSGraph.Chart.8 \s EMBED MSGraph.Chart.8 \s The graph of y = f(-x) is gotten by reflecting the graph of y = f(x) about the y-axis Ex. 12 y = f(x); y = -f(x) + 2 EMBED MSGraph.Chart.8 \s EMBED MSGraph.Chart.8 \s Ex. 13 y = f(-x); y = f(-x) + 3 EMBED MSGraph.Chart.8 \s EMBED MSGraph.Chart.8 \s Vertical Stretching/Shrinking: y = af(x) If a > 1, stretch graph vertically by a factor of a If 0 < a < 1, shrink graph vertically by a factor of a Ex. 14 y = -3x2 EMBED MSGraph.Chart.8 \s Ex. 15 y = .5x2 EMBED MSGraph.Chart.8 \s Ex. 16 f(x) = -3|x – 2| + 4 EMBED MSGraph.Chart.8 \s Horizontal Stretching/Shrinking y = f(ax) If a > 1, shrink graph horizontally by a factor of 1/a If 0 < a < 1, stretch graph horizontally by a factor of 1/a An even function is a function in which f(-x) = f(x). They are symmetrical with respect to the y-axis Ex. 15 Prove f(x) is an even function if f(x) = x4 – x2 + 1 f(-x) = (-x)4 –(-x)2 + 1 = x4 – x2 + 1 f(-2) = -24 –(-2)2 + 1 = 16 – 4 + 1 = 13 f(2) = 24 –(2)2 + 1 = 16 – 4 + 1 = 13 -Therefore, f(x) = f(-x); It is even An odd function is a function in which f(-x) = -f(x); Symmetrical w/ respect to origin Ex. 16 f(x) = x2 – x3 f(-2) = (-2)2 –(-2)3 f(2) = 22 - 23 = 4 –(-8) = 4 – 8 = -4 = 4 + 8 = 12 -Not an odd function