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To determine which graph represents the line [tex]\( y = 3x - 2 \)[/tex], we'll examine key characteristics of the line based on its equation. We can identify important features such as the slope, y-intercept, and the general behavior of the line. Here’s a step-by-step approach:
1. Identify the y-intercept:
- The equation of the line 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]\( y = 3x - 2 \)[/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. Identify the slope:
- The slope ([tex]\( m \)[/tex]) of the line is [tex]\( 3 \)[/tex].
- This means for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 3 units.
- So the line rises steeply.
3. Plot points using the y-intercept and slope:
- Starting from the y-intercept [tex]\( (0, -2) \)[/tex]:
- If [tex]\( x = 1 \)[/tex], then [tex]\( y = 3(1) - 2 = 3 - 2 = 1 \)[/tex]. This gives the point [tex]\((1, 1)\)[/tex].
- If [tex]\( x = -1 \)[/tex], then [tex]\( y = 3(-1) - 2 = -3 - 2 = -5 \)[/tex]. This gives the point [tex]\((-1, -5)\)[/tex].
4. Visualize the line:
- Draw a line passing through the points [tex]\((0, -2)\)[/tex], [tex]\((1, 1)\)[/tex], and [tex]\((-1, -5)\)[/tex].
- The line should be straight, passing through these points, and rising from left to right.
Given these steps, look for the graph that matches the following criteria:
- Crosses the y-axis at [tex]\( (0, -2) \)[/tex].
- Passes through the given points and maintains a slope of 3 (steep incline).
Examine both options (A and B) and check:
- Option A: Check if it crosses the y-axis at [tex]\( (0, -2) \)[/tex] and has a slope of 3.
- Option B: Check if it crosses the y-axis at [tex]\( (0, -2) \)[/tex] and has a slope of 3.
The correct graph will satisfy both conditions.
Ensure that the graph chosen matches these detailed characteristics precisely. If one of the options clearly shows this behavior, that will be the correct graph representing the line [tex]\( y = 3x - 2 \)[/tex].
1. Identify the y-intercept:
- The equation of the line 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]\( y = 3x - 2 \)[/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. Identify the slope:
- The slope ([tex]\( m \)[/tex]) of the line is [tex]\( 3 \)[/tex].
- This means for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 3 units.
- So the line rises steeply.
3. Plot points using the y-intercept and slope:
- Starting from the y-intercept [tex]\( (0, -2) \)[/tex]:
- If [tex]\( x = 1 \)[/tex], then [tex]\( y = 3(1) - 2 = 3 - 2 = 1 \)[/tex]. This gives the point [tex]\((1, 1)\)[/tex].
- If [tex]\( x = -1 \)[/tex], then [tex]\( y = 3(-1) - 2 = -3 - 2 = -5 \)[/tex]. This gives the point [tex]\((-1, -5)\)[/tex].
4. Visualize the line:
- Draw a line passing through the points [tex]\((0, -2)\)[/tex], [tex]\((1, 1)\)[/tex], and [tex]\((-1, -5)\)[/tex].
- The line should be straight, passing through these points, and rising from left to right.
Given these steps, look for the graph that matches the following criteria:
- Crosses the y-axis at [tex]\( (0, -2) \)[/tex].
- Passes through the given points and maintains a slope of 3 (steep incline).
Examine both options (A and B) and check:
- Option A: Check if it crosses the y-axis at [tex]\( (0, -2) \)[/tex] and has a slope of 3.
- Option B: Check if it crosses the y-axis at [tex]\( (0, -2) \)[/tex] and has a slope of 3.
The correct graph will satisfy both conditions.
Ensure that the graph chosen matches these detailed characteristics precisely. If one of the options clearly shows this behavior, that will be the correct graph representing the line [tex]\( y = 3x - 2 \)[/tex].
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