Given a graph of a quadratic polynomial, show that the chord joining any two points with and is parallel to the tangent line at the midpoint
Proof. First, we note that the property of two lines being parallel is the same as the property of the two lines having the same slope.
For any polynomial of degree 2, we may write,
Thus, the slope of the tangent line at the point is
Next, the slope of the chord joining and is given by the difference quotient: