Maria Gaetana Agnesi was a famous 18th-century mathematician who published the first complete book on differential and integral calculus. In that book, she explained a famous function that would later receive the name of "the witch of Agnesi".

1. Here we have three Agnesi's curves. What characteristics do they have in common? Describe them.

2. The functions represented have the following formulas (from top to bottom) (g is red, f is blue, and h is green).

$$\begin{array}{l} \displaystyle g : y = \frac{216}{x^2+36}\\[1em] \displaystyle f : y = \frac{125}{x^2+25}\\[1em] \displaystyle h : y = \frac{27}{x^2+9} \end{array}$$

Can you find the general algebraic expression for the Agnesi's curve?

Tip: Pay attention to the pairs of numbers that appear in each function: 216 and 36, 125 and 5 and 9 and 27. What do they have in common?

3. Here we have represented three Agnesi's curves. Calculate the algebraic expression of the function for a = 4 and represent it with the others.