The rule for a linear function can be written in slope-intercept form, which is y = mx + b, where m is the slope of the line and b is the y-intercept (the point where the line crosses the y-axis). In the image, the line passes through the points (0, 0) and (4, 4). The slope of the line is therefore 4/4 = 1. The y-intercept is also 0, since the line passes through the point (0, 0).Therefore, the rule for the linear function in the image is y = x.
Enter the "slope" and "y-intercept," our trusty sidekicks. Slope, symbolized by "m," reveals how steeply the line climbs as we move right. Imagine it as the line's tilt. The y-intercept, "b," is where the line and the y-axis shake hands.
Armed with the slope-intercept formula (y = mx + b), we can transform the visual clues into a written rule. But first, we need to decode the graph's whispers.
Notice how the line passes through (0, 0) and (4, 4)? These points are like GPS coordinates, telling us valuable intel. (0, 0) hints that the line hugs the origin, meaning b = 0. Tracing from (4, 4) down to the x-axis, we see the line moves 4 units left and 4 units down. This reveals the slope, m = 4/4 = 1.
Plugging our findings into the formula, we unveil the rule:
y = x + 0
This seemingly simple equation holds the line's essence. For every unit we move right (x), the line dips down by 1 unit (y). And since b = 0, it starts from the origin.