To graph the line given by the equation $y = -\frac{1}{2}x + 3$ using the slope and y-intercept, follow these steps:

Step 1: Identify the y-intercept

The y-intercept is the point where the line crosses the y-axis. From the equation, $y = -\frac{1}{2}x + 3$, the y-intercept is 3. This corresponds to the point $(0, 3)$.

Step 2: Use the slope to find another point

The slope $m = -\frac{1}{2}$ tells you how the line changes as you move along it:

• The numerator (-1) indicates a vertical change (downward because of the negative sign).
• The denominator (2) indicates a horizontal change (to the right).

Starting at $(0, 3)$:

1. Move down 1 unit and right 2 units to find the next point: $(2, 2)$.
2. Repeat this process to find additional points, such as $(4, 1)$.

Step 3: Plot the points and draw the line

Plot the points $(0, 3)$, $(2, 2)$, and $(4, 1)$ on the graph. Draw a straight line passing through these points.

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

Define the line equation y = -1/2 * x + 3

x = np.linspace(-10, 10, 100) # x values for the graph y = -0.5 * x + 3 # corresponding y values

Create the plot

plt.figure(figsize=(8, 6)) plt.plot(x, y, label=r"$y = -\frac{1}{2}x + 3$", color="blue") plt.axhline(0, color='black', linewidth=0.5) plt.axvline(0, color='black', linewidth=0.5)

Highlight points (0, 3), (2, 2), and (4, 1)

points = [(0, 3), (2, 2), (4, 1)] for point in points: plt.scatter(*point, color="red", zorder=5) plt.text(point[0] + 0.3, point[1], f"{point}", fontsize=10)

Add labels and grid

plt.title("Graph of y = -1/2x + 3") plt.xlabel("x-axis") plt.ylabel("y-axis") plt.grid(True, which='both', linestyle='--', linewidth=0.5) plt.axhline(0, color="black",linewidth=0.5, linestyle="--") plt.axvline(0, color="black",linewidth=0.5, linestyle="--") plt.legend() plt.show()Here is the graph of the line $y = -\frac{1}{2}x + 3$. The red points represent key points calculated from the slope and y-intercept:

• $(0, 3)$ (y-intercept)
• $(2, 2)$
• $(4, 1)$

Feel free to ask about this graph or any other details!

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Related Questions:

1. How do you determine the slope from a linear equation in standard form?
2. Can you explain how to find x-intercepts of this line?
3. How would the graph change if the slope were positive?
4. What does the slope of this line represent geometrically?
5. How do you graph lines with fractional y-intercepts?

Tip:

Always label your axes and include key points when plotting graphs for better clarity and understanding.