To graph the line given by the equation $y = -4x$, follow these steps:

1. Identify the slope and y-intercept:

   • The equation is already in slope-intercept form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
   • The slope $m = -4$, which means the line goes down 4 units for every 1 unit it moves to the right.
   • The y-intercept $b = 0$, which means the line crosses the y-axis at $(0, 0)$.

2. Plot the y-intercept:

   • Start by plotting the point $(0, 0)$ on the graph, where the line crosses the y-axis.

3. Use the slope to plot another point:

   • From $(0, 0)$, move right 1 unit (positive x-direction) and then down 4 units (because the slope is -4).
   • This brings you to the point $(1, -4)$. Plot this point.

4. Draw the line:

   • Connect the points $(0, 0)$ and $(1, -4)$ with a straight line. The line will continue infinitely in both directions.

Let me plot this for you. matplotlib.pyplot as plt import numpy as np

Define the function y = -4x

x = np.linspace(-10, 10, 400) y = -4 * x

Create the plot

plt.figure(figsize=(6,6)) plt.plot(x, y, label=r'$y = -4x$', color='blue') plt.scatter([0, 1], [0, -4], color='red') # Mark the points (0, 0) and (1, -4) plt.text(0, 0, '(0, 0)', fontsize=12, verticalalignment='bottom') plt.text(1, -4, '(1, -4)', fontsize=12, verticalalignment='top')

Labels and title

plt.axhline(0, color='black',linewidth=1) plt.axvline(0, color='black',linewidth=1) plt.title("Graph of y = -4x") plt.xlabel("x") plt.ylabel("y") plt.grid(True)

Show plot

plt.legend() plt.xlim(-10, 10) plt.ylim(-40, 40) plt.show()Here is the graph of the line $y = -4x$. The red points represent the y-intercept at $(0, 0)$ and another point on the line at $(1, -4)$.

Would you like further details on the graphing process or have any questions about this equation?

Further questions you might find interesting:

1. How would the graph change if the equation was $y = 4x$?
2. What happens to the graph if the slope was 0 (i.e., $y = 0$)?
3. How do you find the equation of a line given two points?
4. How does the graph of $y = -4x + 2$ compare to this one?
5. What is the meaning of slope in a real-world context?

Tip:

When graphing a linear equation, always start by plotting the y-intercept, then use the slope to find another point on the line. This makes drawing the line much easier!