 |
|
Note that all the curves go through the point (1,1). In this picture, the red curves are:
- The horizontal line y=1
- The vertical line x=1
- The diagonal line y=x
- The curve y=1/x
The other four colors will help us distinguish between 4 clumps of curves:
- Gold: functions with exponent greater than one. The bigger the exponent, the closer the curve is to the line
x=1.
Among the functions pictured here, the gold curve closest to the vertical line is
 . The gold curve farthest from the
vertical (that is,
closest to the diagonal line  ), is the familiar parabola,  . You may have
already known that functions in this category get steeper and steeper as they climb forever upwards. Notice how the exponent affects the rate of
growth: the bigger the exponent, the faster it grows (which means it is more nearly vertical). What do you think about the way these functions sag
into that triangle between (0,0), (1,0), and (1,1)?
- Purple: functions with a positive exponent less than one. The one closest to is the
rightward-opening
``half-parabola''
 . Like the gold curves, these grow forever, exceeding all bounds. However, all of these
get closer and closer to horizontal as they go. So, the rate of growth eventually becomes very small.
- Green: functions with a negative exponent greater than -1. Notice how these functions shrink forever as we read left-to-right;
they approach the
x-axis. As we read right-to-left, they abruptly fly upwards as we approach the y-axis, which is a vertical asymptote for all of them.
- Blue: functions with an exponent less than -1. These behave a lot like the green ones, although you can see that they fill up a
different part of
the plane.
Tip your head to the right 45 degrees, and notice how everything seems nicely symmetric about the line y=x. The gold curves are the
mirror images of the purple ones, while the greens are the mirror images of the blues. Do you know why that happens?
|
, for various values of
n.
