91Ó°ÊÓ

The vertex form of a quadratic function is a specific way of writing the equation of a parabola that makes it easy to identify important features. It is expressed as \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola. The vertex is a crucial point on the graph as it represents the peak or the trough of the parabola, depending on its orientation.

In the given exercise, the quadratic function is \(j(x) = -(x-4)^2 + 7\). By seeing the function in this format, we know right away that the vertex is at the point \( (4, 7) \).

Another important aspect of vertex form is the coefficient \(a\). It indicates the direction in which the parabola opens. If \(a\) is positive, the parabola opens upwards. If \(a\) is negative, as in our case where \(a = -1\), the parabola opens downwards. Knowing this is key when we talk about finding maximum or minimum values of the function.

A parabola is the graph of a quadratic function and is a U-shaped curve that can open either up or down. Understanding the properties of a parabola is essential in analyzing functions defined by quadratic equations.

The symmetrical nature of a parabola allows us to determine its behavior easily. The vertex, as mentioned, is the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. The line of symmetry runs vertically through the vertex, ensuring that both halves of the parabola mirror each other.

In our example, the parabola opens downwards because the coefficient of the squared term, \(a = -1\), is negative. It indicates that the curve "frowns" rather than "smiles." Consequently, the vertex represents the maximum point of the parabola, not the minimum.

Quadratic functions can have either a maximum or a minimum value, but never both. This depends predominantly on the direction in which the parabola opens.

If the parabola opens upwards, the vertex will be the lowest point on the graph, thus representing a minimum value. Conversely, if it opens downwards, the vertex is the highest point, representing a maximum value.

• If \(a > 0\), the function's vertex is a minimum.
• If \(a < 0\), the function's vertex is a maximum.

In the exercise, since \(a = -1\), the parabola opens downwards. Therefore, the function does not have a minimum value, because the curve extends towards negative infinity, without bound. Instead, the vertex at \((4, 7)\) provides the maximum value of the function, with \(j(4) = 7\).

Understanding these concepts helps in determining the range of the function and predicting how it behaves graphically. Thus, recognizing how the signs and coefficients in the equation affect the parabola’s orientation and consequent maximum or minimum values is crucial for analyzing quadratic functions.