Chapter 2.1, Problem 88E

Solution

To Prove: The axis of the graph of the quadratic function $f\left(x\right)=a{x}^2+bx+c$ is $x=\left(\frac{x_1+x_2}{2}\right)$ where $x_1$ and $x_2$ are the zeros of the function $f\left(x\right)$ .

Given information: $x_1$ and $x_2$ are the real numbers and the zeros of the following quadratic function:

  $f\left(x\right)=a{x}^2+bx+c$

Proof:

The quadratic equation is $f\left(x\right)=a{x}^2+bx+c$ and the roots of the quadratic function are the zeros of the equation:

Since the graph of quadratic function is a parabola, parallel to the y-axis, with zeros of the equation at the following points:

  $\left(x_1,0\right)$ and $\left(x_2,0\right)$ .

It follows that the axis of the graph is parallel to the y-axis and passes through the point half way between the zeros:

  $x=\left(\frac{x_1+x_2}{2}\right)$