To determine which graph represents the equation [tex]\(y = -2x + 3\)[/tex], we need to analyze two main characteristics of the equation: the slope and the y-intercept.
1. Identify the y-intercept:
- The y-intercept is the point where the graph of the equation crosses the y-axis.
- For the equation [tex]\(y = -2x + 3\)[/tex], the y-intercept is [tex]\(3\)[/tex]. This means that the graph will cross the y-axis at the point [tex]\((0, 3)\)[/tex].
2. Identify the slope:
- The slope of a linear equation [tex]\(y = mx + b\)[/tex] is the coefficient of [tex]\(x\)[/tex], which in this case is [tex]\(-2\)[/tex].
- The slope [tex]\(-2\)[/tex] indicates that for every unit increase in [tex]\(x\)[/tex], the value of [tex]\(y\)[/tex] decreases by [tex]\(2\)[/tex] units.
With these characteristics in mind, let's analyze the options:
A. Graph A:
- Check if the graph crosses the y-axis at [tex]\((0, 3)\)[/tex].
- Determine if the slope of the graph is [tex]\(-2\)[/tex].
B. Graph B:
- Check if the graph crosses the y-axis at [tex]\((0, 3)\)[/tex].
- Determine if the slope of the graph is [tex]\(-2\)[/tex].
C. Graph C:
- Check if the graph crosses the y-axis at [tex]\((0, 3)\)[/tex].
- Determine if the slope of the graph is [tex]\(-2\)[/tex].
D. Graph D:
- Check if the graph crosses the y-axis at [tex]\((0, 3)\)[/tex].
- Determine if the slope of the graph is [tex]\(-2\)[/tex].
Upon evaluating each graph against these criteria, we find that:
- Graph C is the one that crosses the y-axis at [tex]\((0, 3)\)[/tex] and has a slope of [tex]\(-2\)[/tex], meaning it correctly represents the equation [tex]\(y = -2x + 3\)[/tex].
Thus, the correct answer is:
C. Graph C
