Graph the function
f ( x )

To graph the linear function $f(x) = 3x + 2$, let's follow these steps:

Step-by-Step Approach:

1. Identify the key points:

   • The function is a straight line, so it has a slope and a y-intercept.
   • The slope is 3, meaning the line rises 3 units for every 1 unit it moves to the right.
   • The y-intercept is 2, meaning the line crosses the y-axis at $(0, 2)$.

2. Plot the y-intercept:

   • Start by plotting the point $(0, 2)$ on the graph, since this is where the line intersects the y-axis.

3. Use the slope to plot another point:

   • From $(0, 2)$, use the slope 3 to move 3 units up (since the slope is positive) and 1 unit to the right. This gives you the point $(1, 5)$.

4. Draw the line:

   • Once you have two points, $(0, 2)$ and $(1, 5)$, draw a straight line through them. This is the graph of $f(x) = 3x + 2$.

Asymptote:

• For linear functions like this one, there are no horizontal asymptotes. The line continues without bound in both directions.

Would you like me to clarify any of these steps, or do you have any specific questions? Here's a set of related questions to explore further:

1. How does changing the slope affect the graph of a linear function?
2. What happens to the graph if we change the y-intercept?
3. How do you identify the slope and y-intercept from the equation of a line?
4. What is the general form of a linear equation?
5. How do transformations like shifts or reflections apply to linear functions?

Tip: To graph any linear function, always begin by plotting the y-intercept and then use the slope to find a second point. This will give you the direction of the line.