Comparison of the Graphs
The functions y = 34x + 12 and y = x represent two linear equations that can be compared through their slopes and y-intercepts.
1. Understanding the Graphs
- y = x: This is the simplest form of a linear equation where the slope (m) is 1. The graph is a straight line that goes through the origin (0,0) with equal amounts of increase in both x and y. Every increase in x by 1 results in an increase in y by 1. This line bisects the first and third quadrants evenly at a 45-degree angle.
- y = 34x + 12: This equation has a slope of 34, which indicates that for every increase of 1 in x, y increases by 34. Additionally, the y-intercept is 12, meaning the line crosses the y-axis at the point (0,12). This creates a steeper line compared to y = x.
2. Visual Comparison
When visualizing these graphs:
- The graph of y = x will appear as a gentle diagonal line starting from the origin and moving upward.
- The graph of y = 34x + 12 will rise much faster. At x = 1, for instance, y will equal 46 (34 * 1 + 12), while y will only equal 1 for the line y = x. This makes the steep line rise significantly faster compared to y = x.
3. Intersections and Parallelism
These two lines will never intersect because they have different slopes. Furthermore, the line of y = 34x + 12 is not parallel to y = x since they have different slopes (1 vs 34).
4. Summary
In summary, the graph of y = 34x + 12 will rise much more steeply than the graph of y = x. This creates a distinct difference in their visual representation, with y = 34x + 12 dominating the vertical increase as it progresses along the x-axis, while y = x maintains a consistent, moderate slope.