The equation you provided is $y = 3x + 8$, which is in slope-intercept form: $y = mx + b$, where:

• $m = 3$ is the slope,
• $b = 8$ is the y-intercept (where the line crosses the y-axis).

Steps to graph by hand:

1. Start with the y-intercept:

   • Plot the point $(0, 8)$ on the y-axis because when $x = 0$, $y = 8$.

2. Use the slope to find another point:

   • The slope is 3, which means "rise over run." A slope of 3 can be written as $\frac{3}{1}$, meaning for every 1 unit you move to the right (positive direction on the x-axis), move 3 units up (positive direction on the y-axis).
   • Starting from $(0, 8)$, move 1 unit to the right (to $x = 1$) and 3 units up (to $y = 11$). Plot this point: $(1, 11)$.

3. Draw the line:

   • Draw a straight line through the points $(0, 8)$ and $(1, 11)$.
   • You can also extend the line to cover more values by plotting additional points using the slope.

Verifying using a graphing utility:

You can input the equation $y = 3x + 8$ into any graphing calculator or online graphing tool (such as Desmos) to verify that the line you plotted matches the graph generated by the utility.

Would you like a more detailed breakdown or any clarifications?

Here are 5 related questions for practice:

1. What would the graph look like if the slope were negative?
2. How would you graph $y = 3x + 8$ if you were given only the slope and a point not on the y-axis?
3. How do you find the slope if you are given two points?
4. How does changing the value of $b$ affect the graph of a linear equation?
5. What is the equation of a line parallel to $y = 3x + 8$?

Tip: The y-intercept helps you quickly plot the starting point of the line on the graph, and the slope tells you how to move from that point.