# The Multiplication Parabola

In the graph below, move points $A$ and $B$ such that they are on opposite sides of the y-axis.  What do you observe?

[iframe http://mathandmultimedia.com/wp-content/geogebraapplets/multiplicationinparabola/parabolamultiplication.html 500 400]

Theorem: Given two points on the function $f(x) = x^2$, whose coordinates are $(a,a^2)$ and $(b, b^2)$ with $a < 0$ and $b > 0$, the y-intercept of the line passing through these points is $-ab$.

Proof:

First, we get the equation of the line passing through $A$ and $B$.

The equation of the line passing through points is $y - y_1 = m (x-x_1)$ where

$m=\displaystyle\frac{y_2-y_1}{x_2-x_1}$

is the slope of the line. Substituting the values, the slope of the line is

$m = \displaystyle\frac{b^2-a^2}{b-a} = b + a$.

Substituting the value of the slope and the coordinates,  we have $y - a^2 = (b+a) (x-a)$. Simplifying the preceding equation, we have $y = bx - ab + ax$.

Second, we get the y-intercept of the line.

To get he y-intercept, we set $x = 0$. That gives us $y = - ab$ which is what we want to prove.

Teaching Note

In many high schools in our country, proofs is emphasized in geometry which is mostly in two-column form. Many students  have very little knowledge about algebraic proof. When they go to the university, they are shocked when they encounter a lot of algebraic proofs in their classes. The activity above is an example of how we can use problems  to let students generalize, and use algebra to show prove statements.