# The Multiplication Parabola

In the graph below, move points and 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 , whose coordinates are and with and , the *y*-intercept of the line passing through these points is .

Proof:

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

The equation of the line passing through points is where

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

.

Substituting the value of the slope and the coordinates, we have . Simplifying the preceding equation, we have .

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

To get he *y*-intercept, we set . That gives us 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.