22-graphing

22-graphing - Math 1a Graphing Fall, 2009 Some questions to...

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Unformatted text preview: Math 1a Graphing Fall, 2009 Some questions to answer while graphing: (1) What is the domain of y = f (x)? (2) Where does the graph of y = f (x) cross the axes? (3) Where is the tangent to y = f (x) vertical or horizontal? (4) Where is the graph of y = f (x) increasing or decreasing? (5) Where are the local maxima or local minima? (6) What is the "end behavior" of y = f (x)? Are there any asymptotes? (7) Where is the graph concave up? Where is it concave down? Where are the inflection points? Graph the following functions by answering the above questions: 1 f (x) = 9x - x3 2 f (x) = x3 + 6x2 + 12x + 4 3 f (x) = x + x2 + 1 1 = x x 4 f (x) = x2 2x -4 5 f (x) = x2 - 9 1/3 6 f (x) = x2 3x -9 7 f (x) = 3x5 - 20x3 8 f (x) = x2 + 1 x3 + 1 = x x Math 1a Graphing The Graphs Fall, 2009 . ... . . . . . . . . . . . . . . .. . .. . . .. . .. 30 . . .. . .. . . .. .. . . . .. .. . . .. . .. . . .. . .. . 20 . .. .. . . .. . .. . . .. . .. . . .. . .. . ( 3, 6 3)max . .. .. . ... .. . 10 . ........................ .. . . ...... .... .. . ....... .. ... . .. . .. .. ... .. . .. . .. ... .. . .. .. . .. ... . .. ... ... .. . ...................................................................................................................................................................... . ........................................................... .... . . . . . .. . ... .. .... . . -5 -4 -3.. -2 -1 .... . 1 2 3 .. 4 5 .. . ... .. .. . .... . ... . .... .. . .... .. . ..... . .... .. . ... .. ........ .......... . ......... .. -10 . . .. . . .. .. . (- 3, -6 3)min . .. . .. . . .. . .. . . .. .. . . .. -20 . .. . . .. . .. . . .. . .. . . .. .. . . . .. .. . . .. . .. -30 . . .. . . . . . . . . . . . . . . . . . . . . 1 f (x) = 9x - x3 = x(3 - x)(3 + x) f (x) = 9 - 3x2 = -3(x2 - 3) f (x) = -6x . ... . . . . . . . . . . . . . . . . . . . . . .. . 30 . . . . . .. . . . . . . .. . .. . . .. . . .. . . . .. 20 . . .... . . . . . . . . . . . . . . . . . . . . . 10 . .. .. .. .. .. .. . .... . (-2 + 4, 0) . . . .... .. (0, 4) . .... .... . . .. . ........................................................................................................................................................................... ... . . . . .. . .... -5 -4 -3 -2 ................... -1 1 2 . . . ............................ . ...... . . (-2, -4) . ....... . . .... . .. .. . .. -10 . . .. . . .. . . . . . . . . . . . . . . . . . . . . . -20 . . . . . . . .. . . . . . . . . . . . . . . . . . . . -30 . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 3 2 f (x) = x3 + 6x2 + 12x + 4 = (x + 2)3 - 4 f (x) = 3x2 + 12x + 12 = 3(x + 2)2 f (x) = 6x + 12 = 6(x + 2) 3 . ... . . . . . .. .. 10 .. .... .. .. ... .. ... .. ... .. ... .. ... .. .. . .. .. ...... . .. ....... .. ... . .. .. ...... . .. .. ....... y = x ... . .. . .. ...... .. ....... . .. 5 . ... . ... ...... .. . .. ....... ... . .. .. ...... . .. . . ....... . .. ....... . . . .. . .... ........................ . ........ . .. . . (1, 2) . . min . . . . . ............................................................................................................................................................................................................... . . . . -6 -4 -2 2 4 6 . . . . . . ......... . .............. ........ . ..... ... . .. .. . .. . ........ ... ...... .. . . .. . .... ....... .. .. .... .. .. .... .. .. ..... .. .. .. ....... (-1, -2)max . . .... .... .. -5 . . .. .... .. .... . . .. .. ....... . .. .... .. .... . .. .... . .... .. .. .... .. .... . .. .... . . . ........................... . .. Vertical Asymptote x = 0 . .. .. . .. .. . .. .. .. . .. .. .. . . .. -10 .. . . . . . . . . . .. ... .. . .. .. . .... . ... ... . ... ... . ... ... . ... ... ... ... . ... ... ... ... . ... ... . ... ... . ... ... . ... ... ... ... . ... ... ... ... . ... ... . ... ... . ... ... . ... ... ... ... . ... ... ... ... . ... ... . ... ... . ... ... ... ... . ... ... . ... ... ... ... ... ... ... ... . ... ... . ... ... . ... ... . ... ... ... ... ... ... ... ... . ... ... . ... ... . ... ... . ... ... ... ... ... ... . ... ... . ... ... . ... ... . ... ... ... ... .. ... .... . ... ... 1 x2 + 1 f (x) = x + = x x 1 x2 - 1 f (x) = 1 - 2 = x x2 1 2 f (x) = +2 3 = 3 x x . . . . . . . . . . . . 4 . . . . . . . . . . . . . . 3 . . Vertical Asymptote . . . . . .. x = -2 .. .. .. 2 .. .. .. .. ... ... 1 .... ..... ....... ........ ......... ........ ....... x ......1 -4 -3 -2 -1 2 3 4 ..................... ..... ............ .... ......... ...... ... -1 .... ... .... .. .. ... .. .. ... .. .. .. -2 .. . .. . .. . Vertical Asymptote . . .. . .. . . . . x=2 . . . . . -3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . -4 . . . . . . . . . . . . . . . . . . . . . . . . . 4 . . . .. .. . .. .. .. . .. .. .. . .. . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................................................................................................................................................................................................................... .................................................................... .......................................................................................... ................................................................ ... . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. ... .... ..... ....... ......... .............. ................ f (x) = 2x x2 - 4 -2(x2 + 4) f (x) = (x2 - 4)2 4x(x2 + 12) f (x) = (x2 - 4)3 ........ ... ........ ........ ....... ....... .... ....... 2 ...... ...... ...... ..... .... .... ... .... ... ... 1 .. .. (3, 0) .. .. ..(-3, 0)POI . POI . . .. . ...... . .. . ........ .. .. . . . . . . . . . . . x . . . . . -4 -3 -2 -1 1 2 3 4 .. .. .. ... .. .... -1 .. . ..... .... ....... ...... . ......... ......... ............. ................-2................................................... ........ ...... ....... ... (0, - 9)min -3 -4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................................................................................................................................................................................... ............................................................................................................................................................................................................................... ... .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . y ............... 4 3 5 f (x) = x2 - 9 f (x) = 2x 1/3 3 (x2 - 9)2/3 -2(x2 + 27) 9 (x2 - 9)5/3 f (x) = . .. .. .. . . .. .. . .. .. . . .. .. . . . .. .. . . . . .. .. . . .. .. . . .. .. . . . .. .. . . . . .. .. . .. .. . . . . 10 . .. .. . . .. .. . . . . . .. .. . .. . . ................... . . . . Vertical Asymptote x = -3 . .. .. . .. .. . . . .. .. . . . . .. . .. . . . . . .. . .. . . .. . .. . 5 . . .. . .. . . .. . .. . . . .. . ... . .. . .. . . . ... . .... . . ... . .......... . ......... . . . ....................... . . . . ............ .... ................................................................................................................................................................... . . . . ..................................................................................................... .. ......... ................................ ..................................... . . . .. .. .. .. .. .. .. ... ... ........ . . . .. ......... . . . .............. .. . . . -6 -4. -2 2 ..... . 4 6 ..... . . . . . . .... . ... . . . . . . . ... . .. . .. . . . .. . .. . .. . Horizontal Asymptote y = 0 . . .. . . .. . .. . .. -5 . .. . .. . .. . .. .. . .. .. . . . .. .. . . . . .. .. . . ................. . . . .. .. Vertical Asymptote x = 3 .. . .. .. .. . . . .. . .. . .. . .. . . . . .. -10 . . .. .. .. . . .. .. . . . .. . .. . . . .. .. .. . .. .. .. . . .. .. . . .. . .. . . .. . .. .. . .. . . . .. . . . 6 3x f (x) = 2 x -9 -3(x2 + 9) f (x) = (x2 - 9)2 6x(x2 + 27) f (x) = (x2 - 9)3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................ .. .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -4 . . . . . . 400 . . . . . . . . . 300 . . . . . . . . . 200 . . . . . . . . . (-2, 64)max 100 . . .. .. ....... (0, 0)POI ... .................. . ... .. .... ......... . ..... .. .. ..... .. ...................................... .. .. .. . . ............. . . .. .. ........ ........ 2 ..... 3 . .. -2 . -1 -3. 1 . ... .. . .... .. . ... . .. .. . . ... (2, -64)min . . -100 . . . . . ... . . (-2, 282)POI ... . . . . . ( 2, -28 2)POI . -200 . . . . . . . . . -300 . . . . . . . . . -400 . . . . . y ............... 4 x 7 f (x) = 3x5 - 20x3 f (x) = 15x4 - 60x2 = 15x2(x + 2)(x - 2) f (x) = 60x3 - 120x = 60x(x2 - 2) . ... . . . . . . . ... . .. . ... .. . . ... .. . . ... . .. . ... . .. . ... . .. . ... . .. . ... .. . . ... .. . . ... .. . ... . . .. ... . .. . ... . .. ... ... .. .. .. ... ... .. .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... ... .. .. ... ... ... ... ... ... .. . ... 10 .. ... ... ... ... .... .... ... . .. .... ... . .. .. .... .... . .. . .... ... 5 . .. ..... 3 . .. . .. ............................ ( 1/2, 4 )min . .... ...... .......... ....... . ....... (-1, 0)POI .... .. . . .......... ..... . . ..... .......................................................................................................................................................................................................................................... . .... . . .... . -6 -4 -2 2 4 6 ... . . .. . .. . . .. -5 .. . .. . . .......................... .. . .. Vertical Asymptote x = 0 .. .. . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . . . . . .. .. . . . . . .. .. . . . .. .. .. . . .. .. .. .. . .. .. . .. .. . .. .. . .. .. . .. .. . .. .. . ... .. ... .. ... .. ... .. ... .. . .. . .. .. .. .. ... ... .. .. .. .. ... ... ... ... . ... .. .. .. .. .. .. .. . ... .. ... .. ... .. ... .. ... . .. .. .. .. . .. .. . . .. . .. .. .. .. .. .. . .. .. 3 3 8 1 x3 + 1 f (x) = x + = x x 2 1 2x3 - 1 f (x) = 2x - 2 = x x2 2 2(x3 + 1) f (x) = 2 + 3 = x x3 ...
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.

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