Example 2.12.  Show that the transformation  ,  usually maps vertical and horizontal lines onto parabolas and use this fact to find the image of the rectangle .  (a) Find the image of the vertical line  .  (b) Find the image of the horizontal line  .  

Explore Solution 2.12.

Enter the function w = f[z] and determine the real and imaginary parts.

(a) Find the image of the vertical line  .  

Hence, the image of the vertical line    is the parabola  .  

(b) Find the image of the horizontal line  .  

Hence, the image of the horizontal line    is the parabola  .  

Use Mathematica to make a graph of the mapping.

(c) 2006 John H. Mathews, Russell W. Howell