91Ó°ÊÓ

A quadratic function is a type of polynomial function that can be expressed in the standard form as:

• \(y = ax^2 + bx + c\)

Here, \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The term \(ax^2\) defines the quadratic nature of the equation, making it a second-degree polynomial. A unique attribute of quadratic functions is their graph, which is a curve called a "parabola." The direction in which this parabola opens depends on the sign of the leading coefficient \(a\):

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

When working with quadratic functions, one aims to determine the domain and range. Generally, the domain of a quadratic function is all real numbers, i.e., \(-\infty < x < \infty\). This is because there are typically no restrictions on the values that \(x\) can take unless otherwise specified by the context.

Real numbers are a vast set of numbers that include all possible numbers along the continuum: both rational and irrational numbers. Rational numbers are those that can be expressed as a ratio or fraction (like 1/2 or 3), while irrational numbers cannot be written as a simple fraction and often include numbers like \(\sqrt{2}\) and \(\pi\).

• The real number line extends indefinitely in both negative and positive directions.
• Real numbers are important because they provide a complete understanding of how we can measure and quantify quantities in real life, including lengths, areas, and times.

In the context of quadratic functions, when we say that the domain is all real numbers, it means that any real x-value can be plugged into the function without resulting in any undefined or non-real scenario. This results in a smooth, continuous curve on the graph, representing the parabola.

The vertex of a parabola is a critical point that represents the "turning point" where the direction of the parabola changes. It can be either a maximum or minimum point, depending on whether the parabola opens downwards or upwards.For a parabola given by the equation \(y = ax^2 + bx + c\):

• The vertex is located at the point \((h, k)\).
• \(h\) can be found using the formula \(h = -\frac{b}{2a}\).
• The value of \(k\) is obtained by substituting \(h\) back into the function, i.e., \(k = f(h)\).

In our example function, \(y = 2x^2 - 5\), the vertex is found using these steps. The \(x\)-coordinate is 0 because the function can be rewritten as \(y = 2(x-0)^2 - 5\). Substituting \(h = 0\) into the equation yields the \(y\)-coordinate as \(-5\). Hence, the vertex is \((0, -5)\). This point also gives us the minimum value of the function since the parabola opens upwards.