The Multiplication Parabola

July 25, 2011 GB High School Algebra, High School Mathematics

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.

One comment equation of a line, quadratic function, slope of a line, y-intercept of a line

The Multiplication Parabola

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