To graph a line with a slope of $-\frac{1}{2}$ and a $y$-intercept of $1$, let’s first understand the equation of the line in slope-intercept form:

$y = mx + b$

where $m$ is the slope and $b$ is the $y$-intercept.

Given Information:

• Slope ($m$) = $-\frac{1}{2}$
• $y$-intercept ($b$) = $1$

Thus, the equation of the line is:

$y = -\frac{1}{2}x + 1$

Steps to Graph

1. Plot the y-intercept: Start by plotting the point $(0, 1)$ on the graph since that’s where the line crosses the $y$-axis.

2. Use the slope to find another point: The slope $-\frac{1}{2}$ means that for every 2 units you move to the right (positive $x$-direction), you move 1 unit down (negative $y$-direction).

   • From the point $(0, 1)$, move 2 units to the right to reach $x = 2$, and then move 1 unit down to reach $y = 0$. This gives the point $(2, 0)$.

3. Draw the line: Use these two points $(0, 1)$ and $(2, 0)$ to draw a straight line across the graph.

Let me know if you would like me to create this graph or if you have any questions about these steps!

Follow-up Questions

1. What would the graph look like if the slope were positive?
2. How does changing the $y$-intercept affect the position of the line?
3. What is the significance of the slope value in terms of steepness and direction?
4. Can we find the equation of a line parallel to this one?
5. How would you find the equation of a line perpendicular to this one?

Tip

When graphing, always start with the $y$-intercept and then use the slope to determine the direction and spacing of the line.