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A quadratic function is a type of polynomial function characterized by its highest exponent of 2. It takes the form:

• Standard form: \( f(x) = ax^2 + bx + c \)
• Vertex form: \( f(x) = a(x-h)^2 + k \)

Quadratic functions graph as parabolas, which are U-shaped curves. The basic properties of a quadratic function include its domain, range, and the direction the parabola opens. The domain for any quadratic function is all real numbers \((-\infty, +\infty)\), because you can input any real value for \(x\). The range depends on the direction of the opening and the vertex.
Quadratic functions can model various real-world phenomena, such as projectile motion or profit maximization in business, making them essential tools in both algebra and applied mathematics. Understanding their properties is key to mastering other advanced mathematical concepts.

The vertex form of a quadratic function is particularly useful for identifying the vertex of the parabola. This form is expressed as:

• \( g(x) = a(x-h)^2 + k \)

Here, \((h, k)\) represents the vertex of the parabola. This makes it easy to determine the vertex by simply reading the coordinates from the function's equation.
In the given function, \( g(x) = (x - 0)^2 - 5 \), the equation is already in vertex form. It reveals that the vertex is at \((0, -5)\). This vertex is a pivotal point as it signifies the lowest or highest point on the graph of the function depending on the parabola's direction.
From the vertex form, not only can you find the vertex easily, but it tells us immediately much about the symmetry and position of the parabola on the Cartesian plane.

The direction in which a parabola opens is determined by the sign and the value of the coefficient \(a\) in the quadratic function. This crucial aspect determines the orientation of the graph:

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

In the exercise, the function \( g(x) = x^2 - 5 \) has \(a = 1\). Because \(1\) is positive, the parabola opens upwards. This means the vertex \((0, -5)\) is the minimum point of the graph!
Understanding the direction helps in visualizing the overall shape of the parabola and aids in finding the range. Since the vertex is the lowest point for an upward-opening parabola, the range is \([-5, +\infty)\). The parabola extends infinitely higher from its vertex, emphasizing its maximum-reaching nature.