Chapter 9.1, Problem 73HP

To determine

To show that the given statement is false.

Answer to Problem 73HP

The statement is false. A downward facing parabola will have its vertex at the highest point, that is the maximum of the graph.

Explanation of Solution

Given information :

The vertex of a parabola is always the minimum if the graph. Prove false with counterexample

The graph of a quadratic equation ( $y=a{x}^2+bx+c$ ) is a parabola. The parabola is symmetric in nature and the point of symmetry is known as the vertex. This vertex lies on the line of symmetry and is symmetric about this line.

The axis of symmetry bisects the parabola into two equal parts. Hence each point on the parabola would have an equal point on the other side of the axis of symmetry

This vertex point shall be:

Highest point (if $a<0$ ) and be called the maximum. Here, the graph opens downwards.

Or, lowest point (if $a>0$ ) and can be called the minimum. Here the graph opens upwards.

In this case, ‘a’ is lesser than 0 hence the graph will have a maximum and will open downwards.

A parabola always points to infinity, either negative or positive.

Hence, if the quadratic equation has $a<0$ , the graph opens downwards. Therefore, it would have a maximum which would be the vertex

Graph of parabola corresponding to quadratic equation

  $y=-x^2-5$

Here, the vertex is at maximum $(0,5)$ .

Hence, the vertex of a parabola is not always at the minimum of the graph.