To determine which graph represents the equation [tex]\( y = \frac{1}{3}x + 2 \)[/tex], we'll analyze some key points and characteristics of the linear equation.
1. Identify the slope and y-intercept:
- The equation [tex]\( y = \frac{1}{3}x + 2 \)[/tex] is in slope-intercept form [tex]\( y = mx + b \)[/tex].
- Here, the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{3} \)[/tex].
- The y-intercept [tex]\( b \)[/tex] is [tex]\( 2 \)[/tex]. This means the line crosses the y-axis at the point [tex]\( (0, 2) \)[/tex].
2. Plot the y-intercept:
- Start by plotting the point [tex]\( (0, 2) \)[/tex] on the graph.
3. Determine another point using the slope:
- The slope [tex]\( \frac{1}{3} \)[/tex] indicates that for every 3 units increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 1 unit.
- Let's use [tex]\( x = 3 \)[/tex] to find another point on the line:
[tex]\[
y = \frac{1}{3} \times 3 + 2 = 1 + 2 = 3
\][/tex]
- So, another point on the line is [tex]\( (3, 3) \)[/tex].
4. Plot the second point:
- Plot the point [tex]\( (3, 3) \)[/tex] on the graph.
5. Draw the line:
- Draw a straight line through the points [tex]\( (0, 2) \)[/tex] and [tex]\( (3, 3) \)[/tex]. This is the graph of the equation [tex]\( y = \frac{1}{3}x + 2 \)[/tex].
To choose the correct graph among options A, B, C, and D, look for the following key features:
- The line should pass through the point [tex]\( (0, 2) \)[/tex].
- The line should also pass through the point [tex]\( (3, 3) \)[/tex].
- The slope of the line should be [tex]\( \frac{1}{3} \)[/tex], meaning it rises 1 unit for every 3 units it runs horizontally.
Option `D` correctly illustrates these points:
- The line crosses the y-axis at [tex]\( (0, 2) \)[/tex].
- Another visible point on the line is [tex]\( (3, 3) \)[/tex], confirming the slope of [tex]\( \frac{1}{3} \)[/tex].
Therefore, the graph that represents the equation [tex]\( y = \frac{1}{3}x + 2 \)[/tex] is Graph D.
