To graph the line with the equation $y = -\frac{4}{3}x + 1$, let's break down the steps:

1. Identify the Slope and Y-Intercept:

   • The equation is in slope-intercept form $y = mx + b$, where $m$ is the slope, and $b$ is the y-intercept.
   • Here, $m = -\frac{4}{3}$ and $b = 1$.
   • This means the line has a slope of $-\frac{4}{3}$ and crosses the y-axis at $(0, 1)$.

2. Plot the Y-Intercept:

   • Start by plotting the point $(0, 1)$ on the y-axis.

3. Use the Slope to Find Another Point:

   • The slope $-\frac{4}{3}$ tells us that for every 3 units we move to the right (positive $x$-direction), we move 4 units down (negative $y$-direction).
   • From $(0, 1)$, move 3 units right to $x = 3$ and 4 units down to $y = -3$. This gives the point $(3, -3)$.

4. Plot the Second Point and Draw the Line:

   • Plot the point $(3, -3)$.
   • Draw a straight line through these two points, extending it in both directions.

This graph represents the line for the equation $y = -\frac{4}{3}x + 1$.

Would you like a detailed explanation of any of these steps?

Here are some related questions to expand on this:

1. How would the graph change if the slope were positive?
2. What happens to the graph if the y-intercept changes?
3. How can we graph a line given a point and a slope?
4. How do we find the x-intercept of a line given its equation?
5. How does changing the slope affect the steepness of the line?

Tip: The slope determines the direction and steepness of a line. A negative slope means the line descends from left to right.