To determine which graph represents the function [tex]\(y = \frac{2}{3}x - 2\)[/tex], we need to analyze the key characteristics of this linear function, specifically the slope and the y-intercept.
1. Identify the Slope and Y-intercept:
- The function [tex]\(y = \frac{2}{3}x - 2\)[/tex] is in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
- Here, [tex]\(m = \frac{2}{3}\)[/tex] and [tex]\(b = -2\)[/tex].
2. Determine the Y-intercept:
- The y-intercept is the point where the line crosses the y-axis (i.e., when [tex]\(x = 0\)[/tex]).
- Plugging in [tex]\(x = 0\)[/tex] gives [tex]\(y = (2/3) \cdot 0 - 2 = -2\)[/tex].
- So, the y-intercept is [tex]\((0, -2)\)[/tex].
3. Determine Another Point:
- To confidently draw the line, we need another point on the line. Let's use [tex]\(x = 3\)[/tex].
- Plugging in [tex]\(x = 3\)[/tex] gives [tex]\(y = \frac{2}{3} \cdot 3 - 2 = 2 - 2 = 0\)[/tex].
- So, another point on the line is [tex]\((3, 0)\)[/tex].
4. Plot the Points and Draw the Line:
- Plot the two points [tex]\((0, -2)\)[/tex] and [tex]\((3, 0)\)[/tex] on the graph.
- Draw a straight line through these points.
By identifying these two points, [tex]\((0, -2)\)[/tex] and [tex]\((3, 0)\)[/tex], you can verify which graph correctly represents the function [tex]\(y = \frac{2}{3}x - 2\)[/tex]. Look for the graph that passes through these points and has the correct slope of [tex]\(\frac{2}{3}\)[/tex].
