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Sagot :
To determine which graph corresponds to the equation [tex]\( y = -3x - 2 \)[/tex], we'll follow these steps:
### Step-by-Step Solution:
1. Understand the Slope-Intercept Form:
The given equation is in the slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
2. Identify the Slope and Y-Intercept:
- The slope [tex]\( m \)[/tex] is [tex]\(-3\)[/tex]. This means that for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 3 units.
- The y-intercept [tex]\( b \)[/tex] is [tex]\(-2\)[/tex]. This means the line crosses the y-axis at [tex]\( (0, -2) \)[/tex].
3. Plot the Y-Intercept:
Start by plotting the y-intercept on the graph.
- This point is [tex]\( (0, -2) \)[/tex].
4. Use the Slope to Find Another Point:
From the y-intercept [tex]\( (0, -2) \)[/tex], use the slope to find another point on the line.
- Since the slope is [tex]\(-3\)[/tex], from [tex]\( (0, -2) \)[/tex], you move 1 unit to the right (positive [tex]\( x \)[/tex]-direction) and 3 units down (negative [tex]\( y \)[/tex]-direction).
- This gives you the second point [tex]\((1, -5)\)[/tex].
5. Draw the Line:
Now draw a straight line through the points [tex]\( (0, -2) \)[/tex] and [tex]\( (1, -5) \)[/tex].
### Verifying with Graphs:
- Graph A:
- Check if the line in Graph A passes through [tex]\( (0, -2) \)[/tex] and follows a slope of [tex]\(-3\)[/tex].
- Graph B:
- Check if the line in Graph B passes through [tex]\( (0, -2) \)[/tex] and follows a slope of [tex]\(-3\)[/tex].
### Conclusion:
- If Graph A shows a line passing through [tex]\( (0, -2) \)[/tex] and [tex]\( (1, -5) \)[/tex] with a slope of [tex]\(-3\)[/tex], then it corresponds to the equation [tex]\( y = -3x - 2 \)[/tex].
- If Graph B shows these characteristics, then Graph B is the correct graph.
The correct graph is the one that features a line passing through the points identified with the right slope (-3) and y-intercept (-2).
### Step-by-Step Solution:
1. Understand the Slope-Intercept Form:
The given equation is in the slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
2. Identify the Slope and Y-Intercept:
- The slope [tex]\( m \)[/tex] is [tex]\(-3\)[/tex]. This means that for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 3 units.
- The y-intercept [tex]\( b \)[/tex] is [tex]\(-2\)[/tex]. This means the line crosses the y-axis at [tex]\( (0, -2) \)[/tex].
3. Plot the Y-Intercept:
Start by plotting the y-intercept on the graph.
- This point is [tex]\( (0, -2) \)[/tex].
4. Use the Slope to Find Another Point:
From the y-intercept [tex]\( (0, -2) \)[/tex], use the slope to find another point on the line.
- Since the slope is [tex]\(-3\)[/tex], from [tex]\( (0, -2) \)[/tex], you move 1 unit to the right (positive [tex]\( x \)[/tex]-direction) and 3 units down (negative [tex]\( y \)[/tex]-direction).
- This gives you the second point [tex]\((1, -5)\)[/tex].
5. Draw the Line:
Now draw a straight line through the points [tex]\( (0, -2) \)[/tex] and [tex]\( (1, -5) \)[/tex].
### Verifying with Graphs:
- Graph A:
- Check if the line in Graph A passes through [tex]\( (0, -2) \)[/tex] and follows a slope of [tex]\(-3\)[/tex].
- Graph B:
- Check if the line in Graph B passes through [tex]\( (0, -2) \)[/tex] and follows a slope of [tex]\(-3\)[/tex].
### Conclusion:
- If Graph A shows a line passing through [tex]\( (0, -2) \)[/tex] and [tex]\( (1, -5) \)[/tex] with a slope of [tex]\(-3\)[/tex], then it corresponds to the equation [tex]\( y = -3x - 2 \)[/tex].
- If Graph B shows these characteristics, then Graph B is the correct graph.
The correct graph is the one that features a line passing through the points identified with the right slope (-3) and y-intercept (-2).
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