91Ó°ÊÓ

The domain of a function refers to all the possible input values (commonly known as x-values) that the function can accept. For quadratic functions, which are functions of the form \(g(x) = ax^{2} + bx + c\), these often consist of all real numbers. Quadratic functions don't have any restrictions on their x-values. There are no square roots involving negative numbers or divisions by zero present in their equations.
Thus, quadratic functions can handle whatever x-value you throw at them.
However, always keep an eye on functions in general as some other function types may include restrictions on their domains.
But for the function \(g(x) = x^{2} - 5\), its domain is indeed all real numbers, often denoted as \((-\infty, +\infty)\).

• No limitations or restrictions on x-values
• Includes every real number
• The function can be calculated for any x

While the domain focuses on the x-values a function can take, the range concerns itself with the y-values that result from those x-values. It determines the set of possible outputs of a function. In the case of quadratic functions like \(g(x) = x^{2} - 5\), the range depends on the direction the parabola opens and its vertex.
Because our function is written in the form \(ax^{2} + bx + c\), with a positive \(a\), the parabola opens upwards. Given this direction, the lowest y-value occurs at the vertex.
For \(g(x) = x^{2} - 5\), the vertex is at the point (0, -5), which is the minimum point of the function.
Therefore, the range of the function will be all y-values that are -5 or larger: \([-5, +\infty)\).

• Starting point: the y-coordinate of the vertex
• Includes all values greater than or equal to -5
• Due to upward opening, y only increases from the vertex point

Parabolas are the U-shaped graphs of quadratic functions. Every quadratic function like \(g(x) = ax^{2} + bx + c\) will sketch onto a plane as a parabola. The direction the parabola opens—whether upwards or downwards—depends entirely on the leading coefficient \(a\).
If \(a\) is positive, as is the case in \(g(x) = x^{2} - 5\), the parabola opens upwards, suggesting that it has a minimum point rather than a maximum.
Understanding parabolas involves getting to know their key features. A vital feature is the vertex, which is either the lowest or highest point on the graph. For our example, this vertex occurs at (0, -5).
Another critical point is the axis of symmetry, a vertical line that passes through the vertex, dividing the parabola into two equal halves.

• Upward-opening parabolas have a minimum vertex
• Vertex and axis of symmetry can be determined from the function
• Parabolas have a symmetrical structure about their axis

Every parabola is thus uniquely shaped by its quadratic coefficients, leading to rich and diverse applications in various fields.