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?

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.