91Ó°ÊÓ

The vertex of a parabola is a crucial point where the function reaches its maximum or minimum value. In a quadratic function of the form \( g(x) = ax^2 + bx + c \), the vertex serves as the peak or the trough of the graph. To find the vertex, we use a simple formula to get its x-coordinate: \ x = -\frac{b}{2a} \.
Here, \ a \ is the coefficient of \ x^2 \, and \ b \ is the coefficient of \ x \.
After finding the x-coordinate, you can substitute it back into the equation to find the y-coordinate, giving you the full vertex point \ (x, y) \. For example, in the function given \ g(x) = -x^2 + 40x + 10 \, our \ a \ is -1 and \ b \ is 40. We calculate the vertex's x-coordinate like so: \ x = -\frac{40}{2(-1)} = 20 \.

A quadratic equation is a type of polynomial equation of the second degree. It generally has the form \ ax^2 + bx + c = 0 \, where:

• \ a \ is the quadratic coefficient (not zero)
• \ b \ is the linear coefficient
• \ c \ is the constant term

.
The solutions to a quadratic equation are often referred to as the roots. These can be found using methods such as factoring, completing the square, or the quadratic formula given by \[ x = -\frac{b \pm \sqrt{b^2-4ac}}{2a} \] . Additionally, the graph of a quadratic function is a parabola, which can open upwards if \ a \ is positive and downwards if \ a \ is negative.

Maximizing a function involves finding the point at which it reaches its highest value. For quadratic functions that open downwards (where \ a < 0 \), this maximum point is at the vertex. To determine this vertex, follow these steps:

• Identify the coefficients \ a, b, \ and \ c \ in the quadratic equation \ ax^2 + bx + c \
• Compute the x-coordinate of the vertex using \ x = -\frac{b}{2a} \
• Substitute this x-coordinate back into the quadratic function to get the maximum value \ g(x) \

.

Let's look at our example \( g(x) = -x^2 + 40x + 10 \.
We found that \( x = 20 \) is the vertex x-coordinate. To find the maximum value, substitute \( x = 20 \) back into the function: \ g(20) = -20^2 + 40(20) + 10 = 410 \.
Hence, the maximum value of \ g(x) \ is \ 410 \ at \ x = 20 \.