Thread: 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?

is taller than if you plot it only on a bounded segment . Also, for each b > 0, the "girth" of at height b, i.e., where and , is smaller than the analogously defined girth of at height b.