Sure, let's break this down step by step.
First, we'll complete the table by calculating the value of [tex]\( y \)[/tex] for each given [tex]\( x \)[/tex] in the function [tex]\( f(x) = |x| - 3 \)[/tex].
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & y=f(-2) = |-2| - 3 = 2 - 3 = -1 \\
\hline
-1 & y=f(-1) = |-1| - 3 = 1 - 3 = -2 \\
\hline
0 & y=f(0) = |0| - 3 = 0 - 3 = -3 \\
\hline
1 & y=f(1) = |1| - 3 = 1 - 3 = -2 \\
\hline
2 & y=f(2) = |2| - 3 = 2 - 3 = -1 \\
\hline
\end{array}
\][/tex]
So, the completed table looks like this:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & -1 \\
\hline
-1 & -2 \\
\hline
0 & -3 \\
\hline
1 & -2 \\
\hline
2 & -1 \\
\hline
\end{array}
\][/tex]
Next, we'll graph the equation by plotting the vertex and another point.
1. Vertex:
The vertex of the absolute value function [tex]\( f(x) = |x| - 3 \)[/tex] is at [tex]\( (0, -3) \)[/tex].
2. Another Point:
Let's use the point [tex]\( (1, -2) \)[/tex], which we obtained from the table.
Now, let's plot these points on the coordinate system.
1. Plot the vertex [tex]\( (0, -3) \)[/tex].
2. Plot the point [tex]\( (1, -2) \)[/tex].
To complete the graph, we can also use symmetry about the y-axis, as the absolute value function is symmetric. This means that the graph will have the same shape on both sides of the y-axis. Using this, we can plot [tex]\( (-1, -2) \)[/tex], [tex]\( (2, -1) \)[/tex], and [tex]\( (-2, -1) \)[/tex].
Connecting these points, the graph of [tex]\( f(x) = |x| - 3 \)[/tex] will form a "V" shape with the vertex at [tex]\( (0, -3) \)[/tex] and opening upwards from the vertex.
