Show a property of chords between points on a quadratic

Given a graph of a quadratic polynomial, show that the chord joining any two points with $x = a$ and $x = b$ is parallel to the tangent line at the midpoint

$x = \frac{a+b}{2}.$

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 $f$ of degree 2, we may write,

$f(x) = c_2 x^2 + c_1 x + c_0 \quad \implies \quad f'(x) = 2c_2 x + c_1.$

Thus, the slope of the tangent line at the point $x = \frac{a+b}{2}$ is

$f' \left( \frac{a+b}{2} \right) = c_2 (b+a) + c_1.$

Next, the slope of the chord joining $(a,f(a))$ and $(b, f(b))$ is given by the difference quotient:

$\begin{align*} \frac{f(b) - f(a)}{b-a} &= \frac{c_2 b^2 + c_1 b + c_0 - c_2 a^2 - c_1 a - c_0}{b-a} \\ &= \frac{c_2 (b^2 - a^2) + c_1 (b-a)}{b-a} \\ &= c_2 (b+a) + c_1. \qquad \blacksquare \end{align*}$

Stumbling Robot

Show a property of chords between points on a quadratic

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