91Ó°ÊÓ

Let's start by understanding what a quadratic function is. A quadratic function is a type of polynomial function where the highest degree of the variable (usually denoted x) is 2. This means the general form of a quadratic function is \(y = ax^2 + bx + c\).
In our exercise, the quadratic function given is \(y = x^2 - 20\), where \(a = 1\), \(b = 0\), and \(c = -20\).
Quadratic functions graph into a shape called a parabola. If \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards. Because our \(a\) is 1, the parabola for \(y = x^2 - 20\) opens upwards.
A few things that characterize parabolas:

• The vertex: The highest or lowest point of the parabola. For \(y = x^2 - 20\), the vertex is at (0, -20).
• The axis of symmetry: Lines that divide the parabola into two mirror-like halves. For our function, this is the y-axis, \(x = 0\).
• The direction: Dictated by the coefficient 'a', which determines if it opens upwards or downwards.

Understanding the domain and range of a quadratic function is essential. The domain of a function is all the possible x-values (inputs) that you can plug into the function. For any quadratic function, including \(y = x^2 - 20\), the domain is all real numbers \((-\infty, \infty)\), because there's no x-value you can't square and subtract 20 from.
The range is the set of all possible y-values (outputs) that the function can take. For \(y = x^2 - 20\), since the parabola opens upwards and the vertex is at (0, -20), the lowest y-value is -20. Thus, the range is all real numbers greater than or equal to -20, which we write as \([-20, \infty)\).
It's easy to remember that for any parabola opening upwards, the range will start from the y-value of the vertex and go to \(+\infty\). In general, make sure to find the vertex first when determining the range.

To graph a quadratic function like \ y = x^2 - 20 \, we need to plot points. This involves choosing some x-values, calculating the corresponding y-values, and then placing these points on a coordinate plane.
You might follow these steps:

• Choose a set of x-values, for example, from -5 to 5.
• Calculate the y-values using the function. For instance, if \(x = -5\), then \(y = (-5)^2 - 20 = 25 - 20 = 5\). Continue this for all chosen x-values.
• Plot each point (x, y) on the graph. In our case: (-5, 5), (-4, -4), ..., (5, 5).
• Connect the points with a smooth curve, forming the parabola.

This method helps visualize the shape of the graph and ensures accuracy. Additionally, using a graphing calculator can help verify your plotted points and the shape of the parabola.