To graph the equation $y = -x - 2$, follow these steps:

Step 1: Identify Key Components

• The equation is in slope-intercept form:
  $y = mx + b$ where:
  • Slope ($m$) = -1
  • Y-intercept ($b$) = -2 (This is the point where the line crosses the y-axis)

Step 2: Plot the Y-Intercept

• The y-intercept is (0, -2). Plot this point on the graph.

Step 3: Use the Slope to Find Another Point

• The slope is -1, meaning that for every 1 step to the right (increase in x), the y-value decreases by 1.
• From (0, -2), move 1 unit right and 1 unit down to get (1, -3).
• Another point: move 1 unit left and 1 unit up to get (-1, -1).

Step 4: Draw the Line

• Connect the points with a straight line extending in both directions.

Here's a visual representation:

I will generate a graph for you. numpy as np import matplotlib.pyplot as plt

Define the function

def linear_function(x): return -x - 2

Generate x values

x_values = np.linspace(-5, 5, 100) y_values = linear_function(x_values)

Plot the function

plt.figure(figsize=(6,6)) plt.plot(x_values, y_values, label=r"$y=-x-2$", color='blue')

Plot key points

plt.scatter(0, -2, color='red', label="Y-Intercept (0,-2)") plt.scatter(1, -3, color='green', label="(1,-3)") plt.scatter(-1, -1, color='purple', label="(-1,-1)")

Labels and title

plt.axhline(0, color='black', linewidth=1) plt.axvline(0, color='black', linewidth=1) plt.grid(True, linestyle="--", linewidth=0.5) plt.xlabel("x-axis") plt.ylabel("y-axis") plt.title("Graph of y = -x - 2") plt.legend() plt.show()Here is the graph of $y = -x -2$. The line passes through the y-intercept (0, -2) and follows the slope of -1. Let me know if you need any further clarification!

Related Questions:

1. How do you determine the x-intercept of $y = -x - 2$?
2. What happens to the graph if the equation changes to $y = -2x - 2$?
3. How can you find the equation of a parallel line to $y = -x -2$?
4. What is the effect of changing the slope in a linear equation?
5. How can this equation be rewritten in standard form?

Tip: The slope tells you how steep the line is. A negative slope means the line slants downward from left to right.