To graph the line given by the equation $y = -\frac{1}{3}x + 4$, we can follow these steps:

Step 1: Identify the slope and y-intercept.

The equation is in the slope-intercept form $y = mx + b$, where:

• $m$ is the slope of the line,
• $b$ is the y-intercept (the value of $y$ when $x = 0$).

For the equation $y = -\frac{1}{3}x + 4$:

• The slope $m = -\frac{1}{3}$,
• The y-intercept $b = 4$.

Step 2: Plot the y-intercept.

The y-intercept is the point where the line crosses the y-axis. Since $b = 4$, the line will cross the y-axis at the point $(0, 4)$. Mark this point on the graph.

Step 3: Use the slope to find another point.

The slope is $-\frac{1}{3}$, which means for every 3 units you move to the right (positive direction along the x-axis), you move 1 unit down (negative direction along the y-axis).

From the point $(0, 4)$, move 3 units to the right (to $x = 3$) and 1 unit down (to $y = 3$). This gives the point $(3, 3)$. Mark this point.

Step 4: Draw the line.

Now that you have two points, $(0, 4)$ and $(3, 3)$, draw a straight line through these points, extending it in both directions.

This will be the graph of the line $y = -\frac{1}{3}x + 4$.

Would you like further details or any clarifications?

Here are 5 related questions for practice:

1. How would the graph change if the slope was $\frac{1}{3}$ instead of $-\frac{1}{3}$?
2. What happens to the graph if the y-intercept is changed to 2?
3. How do you determine the slope from a graph if only the points are given?
4. What is the equation of a line parallel to this one but passing through $(2, -1)$?
5. How would you write the equation of a line with slope 2 and y-intercept -3?

Tip: Remember, the slope tells you how steep the line is and the direction it moves (up or down).