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A parabola is a U-shaped curve that you get when you graph a quadratic function. In our example, the function is given by \(f(x) = -x^2 - 1\). This specific shape helps us understand various properties of the quadratic function. Because of the negative coefficient of \(x^2\), our parabola opens downwards. Parabolas have some key components: the vertex, which is the highest point if it opens downwards and the lowest point if it opens upwards, the axis of symmetry, which is a vertical line that runs through the vertex and splits the parabola into two mirror images, and the direction in which it opens.

A coordinate plane is a two-dimensional surface where you can plot points, lines, and curves. It is defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Each point in the plane is identified by an ordered pair of numbers (x, y). The first number is the x-coordinate, which tells how far to move left or right from the origin (0, 0). The second number is the y-coordinate, which tells how far to move up or down. For our example, we'll be plotting points that we calculate using the function \(f(x) = -x^2 -1\). The coordinate plane allows us to visually represent these points and better understand the behavior of the quadratic function. We start by creating axes and marking appropriate scales for both x and y values.

A table of values helps us to systematically calculate and organize the points that we will later plot on the coordinate plane. To create a table of values for the function \(f(x) = -x^2 - 1\), follow these steps:

• Choose several x-values (both positive and negative) to get a good shape of the parabola.
• Plug each x-value into the function to find the corresponding y-value.
• Record the pairs of (x, y) values in a table for easy reference.

For example, if we choose x-values of -2, -1, 0, 1, and 2, we can calculate corresponding y-values:

• For x = -2, \(f(-2) = -(-2)^2 - 1 = -5\)
• For x = -1, \(f(-1) = -(-1)^2 - 1 = -2\)
• For 0, \(f(0) = -(0)^2 - 1 = -1\)
• For 1, \(f(1) = -(1)^2 - 1 = -2\)
• For 2, \(f(2) = -(2)^2 - 1 = -5\)

Plotting points is the basic way to start creating the graph of a function. Here’s how to plot the points you've calculated in the table of values:

• Take each pair of (x, y) values from the table of values.
• Locate the x-coordinate on the x-axis.
• Move vertically to reach the y-coordinate.
• Mark the point where x and y meet on the coordinate plane.

For our example, plot the points (-2, -5), (-1, -2), (0, -1), (1, -2), and (2, -5). Once all points are plotted, draw a smooth curve that connects them, creating the graph of the quadratic function \(f(x) = -x^2 - 1\). This curve should form a downward-opening parabola, reflecting the nature of the function.