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Math Help - graph of functions

  1. #1
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    graph of functions

    Function Transformations / Translations: Additional Rules
    It mentions "The parabola for 2x2 grows twice as fast as x2, so its graph is tall and skinny. On the other hand, the parabola for the function ( 1/2 )x2 grows only half as fast, so its graph is short and fat.".

    How come the parabola for 2x2is tall and skinny? Aren't both the parabola for 2x2 and ( 1/2 )x2 of the same "height"? (Are they both go to infinity?) So why is the parabola for 2x2 tall and the parabola for( 1/2 )x2 short?
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  2. #2
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    Re: graph of functions

    2x^2 is taller than (1/2)x^2 if you plot it only on a bounded segment [-a, a]. Also, for each b > 0, the "girth" of 2x^2 at height b, i.e., |x_2(b)-x_1(b)| where x_1(b)\ne x_2(b) and 2(x_1(b))^2=2(x_2(b))^2=b, is smaller than the analogously defined girth of (1/2)x^2 at height b.
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