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?

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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.


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


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.

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