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