91Ó°ÊÓ

A quadratic function is an equation that can be written in the form of \( y = Ax^2 + Bx + C \), where \( A \), \( B \), and \( C \) are constants, and \( A eq 0 \). This kind of function graphs as a parabola—a U-shaped curve. The behavior of the parabola depends on the sign of \( A \):

• If \( A > 0 \), the parabola opens upwards.
• If \( A < 0 \), it opens downwards.

The quadratic function is fundamental in calculus optimization because it models many physical phenomena where finding the maximum or minimum values is crucial.

In this exercise, the given function \( y = (x-a)^2 + (x-b)^2 \) was expanded to show it as a quadratic equation \( y = 2x^2 - 2(a+b)x + (a^2 + b^2) \). This expansion reveals that it's a quadratic function with \( A = 2 \), meaning the parabola opens upwards and thus has a minimum point.

The vertex formula is a key tool in finding the minimum or maximum point of a quadratic function. The x-coordinate of the vertex, for a given quadratic equation \( y = Ax^2 + Bx + C \), is determined by the formula:

• \( x = -\frac{B}{2A} \)

This specific formula helps pinpoint where the quadratic function turns, either at a minimum if the parabola opens upwards or a maximum if it opens downwards.

In the example provided, the expression simplified to \( y = 2x^2 - 2(a+b)x + (a^2 + b^2) \), we have \( B = -2(a+b) \) and \( A = 2 \). Using the vertex formula allows us to calculate:
\[ x = -\frac{-2(a+b)}{2 \times 2} = \frac{a+b}{2} \]
This x-value corresponds to the turning point of the function, which is the minimum because the parabola opens upwards as confirmed by \( A = 2 \).

Identifying the minimum point of a parabola is an essential aspect of solving optimization problems in calculus. The parabola's vertex represents the point of either the maximum or minimum value. Since the sign of \( A \) in the quadratic function determines the parabola's direction:

• If \( A > 0 \): The vertex represents the minimum point.
• If \( A < 0 \): The vertex represents the maximum point.

In this exercise, after expansion, the quadratic equation \( y = 2x^2 - 2(a+b)x + (a^2 + b^2) \) has a positive \( A \) coefficient. This positive value means the parabola opens upwards, making the vertex a minimum point.

The vertex, calculated at \( x = \frac{a+b}{2} \), gives us the precise x-coordinate where the minimum y-value occurs. This result illustrates how the vertex formula and understanding the parabola's direction are vital for locating the point of optimization in a given equation.