To graph the equation $y = -4$ for the given values of $x$ $(-3, -2, -1, 0, 1, 2, 3)$, follow these steps:

1. Recognize that $y = -4$ is a horizontal line where the $y$-value is always -4, regardless of the $x$-value.

2. Plot the points for the given $x$-values with $y = -4$:

   • For $x = -3$, $y = -4$
   • For $x = -2$, $y = -4$
   • For $x = -1$, $y = -4$
   • For $x = 0$, $y = -4$
   • For $x = 1$, $y = -4$
   • For $x = 2$, $y = -4$
   • For $x = 3$, $y = -4$

3. Draw a horizontal line passing through all these points on the coordinate plane.

Here is the graph:

x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ y & -4 & -4 & -4 & -4 & -4 & -4 & -4 \\ \end{array}$$ Now let's visualize this on a coordinate plane: \[ \begin{tikzpicture} \begin{axis}[ axis lines = middle, xlabel = $$x$$, ylabel = $$y$$, ymin = -5, ymax = -3, xmin = -4, xmax = 4, xtick={-3, -2, -1, 0, 1, 2, 3}, ytick={-4}, grid = major, width=10cm, height=6cm ] \addplot[ color=blue, mark=*, only marks, domain=-3:3 ] {-4}; \addplot[ color=blue, samples=100, domain=-3:3 ] {-4}; \end{axis} \end{tikzpicture} \] ### Would you like more details or have any questions? ### Further Exploration Questions: 1. How would the graph look if the equation were $$y = 4$$ instead? 2. What is the slope of the line $$y = -4$$? 3. How can we express the horizontal line equation in slope-intercept form? 4. What is the y-intercept of the line $$y = -4$$? 5. How does the line $$y = -4$$ compare to the line $$x = -4$$? 6. If you shift the line $$y = -4$$ up by 5 units, what will the new equation be? 7. What will be the coordinates of the points if $$x$$ ranges from -5 to 5 for the equation $$y = -4$$? 8. How would the graph change if the equation were $$y = -4 + x$$? ### Tip: When graphing horizontal lines, remember that the y-value remains constant across all x-values.