Vertex of a quadratic curve
We can rewrite a quadratic function
y = a x2 + b x + c
in the form
This has the form constant + a times square. Thus if a is positive, the
function has a minimum, since we are always adding a value >= 0 to the constant.
Likewise, if a is negative, then the function has a maximum. The point
on the graph where the maximum or minimum occurs is called the vertex.
We can find its x-coordinate from the fact that the square term in the
equation shown above must be 0 at the vertex. Hence, at the vertex,
x = -b/(2a).
Example
For the curve y = - x2 + 3 x + 1, we have a = -1 and b = 3. Thus
the curve has a maximum because a < 0, and the x-coordinate at the vertex
is -b/(2a) = 3/2. The graph is shown below:
Tutorial applet
The applet poses random problems about this topic.