To determine which graph represents the equation [tex]\( y = \frac{1}{3}x + 2 \)[/tex], we need to analyze the components of the given linear equation:
1. Slope: The slope of the line is [tex]\( \frac{1}{3} \)[/tex]. This means for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by [tex]\( \frac{1}{3} \)[/tex] units. The line is relatively shallow, not steep.
2. Y-Intercept: The y-intercept is 2. This means when [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex]. The line crosses the y-axis at the point (0, 2).
To find the correct graph, ensure it meets these criteria:
- The line should pass through the y-axis at the point (0, 2).
- The slope should reflect a rise of 1 unit for every 3 units it runs horizontally.
### Step-by-Step Verification:
1. Identify the Y-Intercept: Check if the line passes through the point (0, 2).
2. Verify the Slope:
- Starting from the y-intercept (0, 2), if you move 3 units to the right (along the x-axis), the line should move up 1 unit (along the y-axis).
- For example, from (0, 2) to (3, 3), where 3 is calculated as [tex]\( 2 + 1 \)[/tex] (since 1 is the change in [tex]\( y \)[/tex] when [tex]\( x \)[/tex] changes by 3).
Given the criteria above, look at the options and see:
- A graph that crosses the y-axis at [tex]\( y = 2 \)[/tex].
- A graph that shows a gentle increase, with a rise of 1 for every run of 3 units.
By analyzing the right graphical representation that meets both criteria, we conclude that the correct graph corresponds to the equation [tex]\( y = \frac{1}{3} x + 2 \)[/tex].
