The task asks to graph the line $y = x + 1$ on a rectangular coordinate system. I'll plot this for you. numpy as np import matplotlib.pyplot as plt

Define the x and y coordinates for the line

x = np.linspace(-10, 10, 400) # x values from -10 to 10 y = x + 1 # y values based on the equation y = x + 1

Create the plot

plt.figure(figsize=(8, 6)) plt.plot(x, y, label=r"$y = x + 1$", color="blue") plt.axhline(0, color="black", linewidth=0.8) # x-axis plt.axvline(0, color="black", linewidth=0.8) # y-axis

Add labels and title

plt.title("Graph of y = x + 1", fontsize=14) plt.xlabel("x", fontsize=12) plt.ylabel("y", fontsize=12) plt.grid(color='gray', linestyle='--', linewidth=0.5) plt.legend(fontsize=12) plt.axis('equal')

Display the plot

plt.show()Here is the graph of the line $y = x + 1$ on the rectangular coordinate system. The line intersects the y-axis at $(0, 1)$ and has a slope of $1$, meaning it rises one unit for every one unit it moves to the right.

Would you like further clarification or have any questions?

Here are five related questions for deeper understanding:

1. How do you determine the slope and y-intercept from a linear equation in slope-intercept form?
2. What is the significance of the slope in a straight line?
3. How can you find where the line $y = x + 1$ intersects the x-axis?
4. What changes in the graph if the equation is $y = x - 2$?
5. How can you graph a line if given two points instead of an equation?

Tip: The slope-intercept form $y = mx + b$ makes it easy to graph a line by starting at the y-intercept $b$ and using the slope $m$ to find additional points.