To determine which of the graphs represents the equation [tex]\(y = -2x + 3\)[/tex], follow these steps:
1. Identify the slope and y-intercept from the equation: The given equation is in the form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
- For the equation [tex]\(y = -2x + 3\)[/tex], the slope ([tex]\(m\)[/tex]) is [tex]\(-2\)[/tex].
- The y-intercept ([tex]\(b\)[/tex]) is [tex]\(3\)[/tex].
2. Understand what these parameters mean:
- Y-intercept: The line crosses the y-axis at [tex]\(y = 3\)[/tex]. This means at [tex]\(x = 0\)[/tex], [tex]\(y\)[/tex] will be [tex]\(3\)[/tex].
- Slope: The slope of [tex]\(-2\)[/tex] means that for every unit increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] decreases by [tex]\(2\)[/tex].
3. Locate the key points:
- When [tex]\(x = 0\)[/tex], [tex]\(y = 3\)[/tex]. This gives us the point [tex]\((0, 3)\)[/tex] which should be on the line.
- Determine another point using the slope. If [tex]\(x = 1\)[/tex], then:
[tex]\[
y = -2(1) + 3 = -2 + 3 = 1 \quad \text{(second point would be at} \ (1,1)\).
\][/tex]
4. Check the behavior of the line: Look for a line that:
- Crosses the y-axis at [tex]\(3\)[/tex] (i.e., the point [tex]\((0, 3)\)[/tex]).
- Slopes downwards (decreases) as it moves to the right.
Finally, examining the graphs:
- Graph A: Check if it has the intercept at [tex]\(3\)[/tex] on the y-axis and the downward slope.
- Graph B: Similar check.
- Graph C: Similar check.
- Graph D: Similar check.
After analyzing the graphs with these criteria in mind, identify the one that meets both conditions.
Given the detailed analysis, the graph that represents the equation [tex]\(y = -2x + 3\)[/tex] will be clear by looking for the above-discussed characteristics in the provided graphs.
