To determine which graph is generated by the given table of values, we need to analyze the relationship between the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values.
Here's a table for reference:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 1 & 2 & 4 & 5 \\
\hline
y & -3 & -3 & -3 & -3 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Solution:
1. Identify the Relationship:
- Observe that the [tex]\( y \)[/tex]-values are all equal to [tex]\(-3\)[/tex], irrespective of the [tex]\( x \)[/tex]-values. This means the value of [tex]\( y \)[/tex] does not change as [tex]\( x \)[/tex] varies.
2. Determine the Type of Line:
- When all the [tex]\( y \)[/tex]-values are the same, it indicates a horizontal line. This is because a horizontal line has a constant [tex]\( y \)[/tex]-value across all [tex]\( x \)[/tex]-values.
3. Identify the Slope:
- For horizontal lines, the slope ([tex]\( m \)[/tex]) is always zero because there is no vertical change as [tex]\( x \)[/tex] changes. Mathematically, [tex]\( m = 0 \)[/tex].
4. Determine the Equation of the Line:
- With a horizontal line, the equation is of the form [tex]\( y = c \)[/tex], where [tex]\( c \)[/tex] is the constant [tex]\( y \)[/tex]-value.
- Given that [tex]\( y \)[/tex] is consistently [tex]\(-3\)[/tex], the equation of the line is [tex]\( y = -3 \)[/tex].
5. Graph Generated:
- To visualize, plot the given [tex]\( x \)[/tex]-values (1, 2, 4, 5) on the horizontal axis, and for each [tex]\( x \)[/tex]-value, plot the [tex]\( y \)[/tex]-value as [tex]\(-3\)[/tex] on the vertical axis.
- Connect these points to form a straight horizontal line passing through [tex]\( y = -3 \)[/tex].
So, the graph generated by the given table of values is a horizontal line at [tex]\( y = -3 \)[/tex].
