To graph the line with a slope of $\frac{1}{2}$ and a $y$-intercept of 6, we can use the slope-intercept form of a linear equation:

Slope-Intercept Form:

$y = mx + b$ where:

• $m$ is the slope of the line,
• $b$ is the $y$-intercept.

Given values:

• Slope $m = \frac{1}{2}$,
• $y$-intercept $b = 6$.

Equation of the Line:

Substitute $m$ and $b$ into the formula: $y = \frac{1}{2}x + 6$

Steps to Graph the Line:

1. Plot the $y$-intercept (0,6): Start by plotting the point where the line crosses the $y$-axis at $(0, 6)$.

2. Use the Slope to Find Another Point: The slope $\frac{1}{2}$ means that for every 1 unit increase in $x$, $y$ increases by $\frac{1}{2}$. Starting from $(0,6)$:

   • Move 1 unit to the right (in the $x$-direction) and 0.5 units up (in the $y$-direction) to get the next point at $(1, 6.5)$.

3. Draw the Line: Connect these points with a straight line, extending it in both directions.

Would you like me to provide a visual graph of this line?

Here are some follow-up questions for further practice:

1. What would be the equation of a line with the same slope but a $y$-intercept of 3?
2. How would the graph change if the slope were negative, say $-\frac{1}{2}$?
3. What is the equation of a line that passes through the origin with a slope of $\frac{1}{2}$?
4. How can you determine the slope and $y$-intercept from any linear equation?
5. How would you graph a line given two points instead of the slope and $y$-intercept?

Tip: Remember that the slope describes the "rise over run," helping you move from point to point on the graph.