The graph of the equation y = mx, where m is greater than 0, represents a straight line through the origin (0, 0) of the Cartesian plane. This line has a positive slope, which means that as the value of x increases, the value of y also increases.
To understand this better, let’s break down the components:
- Positive Slope (m): The slope (m) determines how steep the line is. When m is positive (e.g., 1, 2, 3), the line rises as it moves from left to right. For instance, if m = 1, the line rises at a 45-degree angle, while a larger value of m like 3 results in a steeper incline.
- Y-Intercept: In the equation y = mx, there is no constant term added, which means the y-intercept is 0. The line crosses the y-axis at the point (0, 0).
- Quadrants: Since the line passes through the origin and rises to the right, it will occupy the first and third quadrants of the Cartesian plane. It will trend upwards in the first quadrant (where both x and y are positive) and downwards in the third quadrant (where both x and y are negative).
Graphically, you can visualize this line starting at the origin and moving towards the right. The steeper the line, the larger the value of m. For example:
- If m = 1, the line will rise left to right at a 45-degree angle.
- If m = 2, the line will be steeper, rising twice as fast.
In summary, the graph of y = mx with m > 0 shows a straight line that begins at the origin and continuously ascends as it moves to the right. This creates a simple yet powerful illustration of positive linear relationships, common in various fields such as mathematics, economics, and the sciences.