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To determine which graph represents the line [tex]\( y = -3x + 2 \)[/tex], we need to analyze the line's key features.
### Step-by-Step Solution:
1. Identify the Slope and y-Intercept:
The given equation 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.
- The slope ([tex]\( m \)[/tex]) is [tex]\(-3\)[/tex]. This means that for every increase of 1 unit in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 3 units.
- The y-intercept ([tex]\( b \)[/tex]) is [tex]\(2\)[/tex]. This is the point where the line crosses the y-axis.
2. Calculate Key Points on the Line:
To find specific points that lie on the line, we can choose a few values for [tex]\( x \)[/tex] and solve for [tex]\( y \)[/tex].
- Point at [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -3(0) + 2 = 2 \][/tex]
So, the point is [tex]\((0, 2)\)[/tex].
- Point at [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -3(1) + 2 = -1 \][/tex]
So, the point is [tex]\((1, -1)\)[/tex].
- Point at [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -3(-1) + 2 = 5 \][/tex]
So, the point is [tex]\((-1, 5)\)[/tex].
3. Plot the Points and Sketch the Line:
- Plot the point [tex]\((0, 2)\)[/tex] — where the line crosses the y-axis.
- Plot the point [tex]\((1, -1)\)[/tex] — when [tex]\( x = 1 \)[/tex], [tex]\( y \)[/tex] is [tex]\(-1\)[/tex].
- Plot the point [tex]\((-1, 5)\)[/tex] — when [tex]\( x = -1 \)[/tex], [tex]\( y \)[/tex] is [tex]\( 5 \)[/tex].
4. Analyze the Characteristics:
- The line through these points should have a negative slope (indicating it slopes downward from left to right).
- The line intercepts the y-axis at [tex]\(2\)[/tex].
- The line decreases by 3 units in [tex]\( y \)[/tex] for every unit increase in [tex]\( x \)[/tex].
By identifying these characteristics and plotting the points correctly, you can then compare them to the provided graphs (A, B, C, D). The correct graph will match these features:
- It will have a y-intercept at [tex]\((0, 2)\)[/tex].
- It will pass through the points [tex]\((1, -1)\)[/tex] and [tex]\((-1, 5)\)[/tex].
- It will slope downward with the slope of [tex]\(-3\)[/tex].
Based on this analysis, match these points and the slope with the graphs given in the options to select the correct one.
### Step-by-Step Solution:
1. Identify the Slope and y-Intercept:
The given equation 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.
- The slope ([tex]\( m \)[/tex]) is [tex]\(-3\)[/tex]. This means that for every increase of 1 unit in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 3 units.
- The y-intercept ([tex]\( b \)[/tex]) is [tex]\(2\)[/tex]. This is the point where the line crosses the y-axis.
2. Calculate Key Points on the Line:
To find specific points that lie on the line, we can choose a few values for [tex]\( x \)[/tex] and solve for [tex]\( y \)[/tex].
- Point at [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -3(0) + 2 = 2 \][/tex]
So, the point is [tex]\((0, 2)\)[/tex].
- Point at [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -3(1) + 2 = -1 \][/tex]
So, the point is [tex]\((1, -1)\)[/tex].
- Point at [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -3(-1) + 2 = 5 \][/tex]
So, the point is [tex]\((-1, 5)\)[/tex].
3. Plot the Points and Sketch the Line:
- Plot the point [tex]\((0, 2)\)[/tex] — where the line crosses the y-axis.
- Plot the point [tex]\((1, -1)\)[/tex] — when [tex]\( x = 1 \)[/tex], [tex]\( y \)[/tex] is [tex]\(-1\)[/tex].
- Plot the point [tex]\((-1, 5)\)[/tex] — when [tex]\( x = -1 \)[/tex], [tex]\( y \)[/tex] is [tex]\( 5 \)[/tex].
4. Analyze the Characteristics:
- The line through these points should have a negative slope (indicating it slopes downward from left to right).
- The line intercepts the y-axis at [tex]\(2\)[/tex].
- The line decreases by 3 units in [tex]\( y \)[/tex] for every unit increase in [tex]\( x \)[/tex].
By identifying these characteristics and plotting the points correctly, you can then compare them to the provided graphs (A, B, C, D). The correct graph will match these features:
- It will have a y-intercept at [tex]\((0, 2)\)[/tex].
- It will pass through the points [tex]\((1, -1)\)[/tex] and [tex]\((-1, 5)\)[/tex].
- It will slope downward with the slope of [tex]\(-3\)[/tex].
Based on this analysis, match these points and the slope with the graphs given in the options to select the correct one.
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