91Ó°ÊÓ

In quadratic functions like the one given, the maximum or minimum value is always located at the vertex of the parabola. Since our function is in the form of a downward-opening parabola (because the coefficient of the \(x^{2}\) term is negative), it will have a maximum value.
The y-coordinate of this vertex represents the maximum value of the function. To find this, we first need to determine the vertex, which we will discuss next. Once we have the x-coordinate of the vertex, we substitute it back into the function to get the maximum value.
This is a useful feature of quadratic functions, as it allows us to find the highest (or lowest) point easily, based on the orientation of the parabola.

Calculating the vertex of a quadratic function is crucial because it tells us where the maximum or minimum value occurs. The vertex formula for a quadratic function \(ax^{2} + bx + c\) is given by: \[x_v = -\frac{b}{2a}\].

• Identify the coefficients: Here, we have \(a = -1\) and \(b = 40\).
• Substitute these into the formula: \[x_v = -\frac{40}{2(-1)} = \frac{40}{2} = 20\].

So, the x-coordinate of the vertex is 20.
This vertex represents the point where the function attains its maximum value (since the parabola opens downwards).

A parabola is the graph of a quadratic function and it can either open upwards (forming a 'U' shape) or downwards (forming an 'n' shape). The direction it opens is determined by the sign of the coefficient of the \(x^{2}\) term:

• If the coefficient is positive, the parabola opens upwards and has a minimum value at its vertex.
• If the coefficient is negative, like in our case (\(g(x)=10+40x-x^{2}\)), the parabola opens downwards and has a maximum value at its vertex.

The vertex is the highest or lowest point of the parabola, and it lies on its axis of symmetry. Understanding the nature of parabolas helps us quickly determine the key characteristics, such as the vertex and the maximum or minimum value of the function.