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Sagot :
Sure! Let's graph the line represented by the equation [tex]\( y = -5x + 2 \)[/tex]. Here's a step-by-step approach:
### Step 1: Understanding the Equation
The equation [tex]\( y = -5x + 2 \)[/tex] is a linear equation in slope-intercept form, [tex]\( y = mx + b \)[/tex], where:
- [tex]\( m \)[/tex] is the slope.
- [tex]\( b \)[/tex] is the y-intercept.
For [tex]\( y = -5x + 2 \)[/tex]:
- The slope [tex]\( m \)[/tex] is [tex]\(-5\)[/tex].
- The y-intercept [tex]\( b \)[/tex] is [tex]\(2\)[/tex].
### Step 2: Plotting the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. This occurs when [tex]\( x = 0 \)[/tex].
1. Substitute [tex]\( x = 0 \)[/tex] in the equation:
[tex]\[ y = -5(0) + 2 = 2 \][/tex]
2. Plot the point [tex]\((0, 2)\)[/tex] on the graph.
### Step 3: Using the Slope to Find Another Point
The slope of [tex]\(-5\)[/tex] means that for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 5 units.
1. From the y-intercept [tex]\((0, 2)\)[/tex], move 1 unit to the right (to [tex]\( x = 1 \)[/tex]).
2. Calculate the corresponding [tex]\( y \)[/tex]-value:
[tex]\[ y = -5(1) + 2 = -3 \][/tex]
3. Plot the point [tex]\((1, -3)\)[/tex].
### Step 4: Drawing the Line
Now that we have at least two points [tex]\((0, 2)\)[/tex] and [tex]\((1, -3)\)[/tex], we can draw the line:
1. Draw a straight line through these points.
2. Extend the line in both directions, beyond the plotted points.
### Step 5: Verifying with More Points (Optional)
If you want to be more thorough, you can plot additional points:
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -5(-1) + 2 = 7 \][/tex]
So, plot the point [tex]\((-1, 7)\)[/tex].
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -5(2) + 2 = -8 \][/tex]
So, plot the point [tex]\((2, -8)\)[/tex].
### Final Graph
You should see that these points line up in a straight line, confirming that your graph is correct. When graphing, make sure to label your axes and units clearly for clarity.
Your graph should look like this:
```
Graph: y = -5x + 2
y
|
10|
9|
8|
7|
6|
5|
4|
3|
2|------o-------
1|
0|----------------------------------------- x
-1|
-2|
-3|
-4|
-5|
-6|
-7|
-8| *
```
- The point at [tex]\( (0, 2) \)[/tex] represents the y-intercept.
- The point at [tex]\( (1, -3) \)[/tex] shows the slope effect.
- Additional points should fit the same straight line.
This completes our graph of the line represented by the equation [tex]\( y = -5x + 2 \)[/tex].
### Step 1: Understanding the Equation
The equation [tex]\( y = -5x + 2 \)[/tex] is a linear equation in slope-intercept form, [tex]\( y = mx + b \)[/tex], where:
- [tex]\( m \)[/tex] is the slope.
- [tex]\( b \)[/tex] is the y-intercept.
For [tex]\( y = -5x + 2 \)[/tex]:
- The slope [tex]\( m \)[/tex] is [tex]\(-5\)[/tex].
- The y-intercept [tex]\( b \)[/tex] is [tex]\(2\)[/tex].
### Step 2: Plotting the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. This occurs when [tex]\( x = 0 \)[/tex].
1. Substitute [tex]\( x = 0 \)[/tex] in the equation:
[tex]\[ y = -5(0) + 2 = 2 \][/tex]
2. Plot the point [tex]\((0, 2)\)[/tex] on the graph.
### Step 3: Using the Slope to Find Another Point
The slope of [tex]\(-5\)[/tex] means that for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 5 units.
1. From the y-intercept [tex]\((0, 2)\)[/tex], move 1 unit to the right (to [tex]\( x = 1 \)[/tex]).
2. Calculate the corresponding [tex]\( y \)[/tex]-value:
[tex]\[ y = -5(1) + 2 = -3 \][/tex]
3. Plot the point [tex]\((1, -3)\)[/tex].
### Step 4: Drawing the Line
Now that we have at least two points [tex]\((0, 2)\)[/tex] and [tex]\((1, -3)\)[/tex], we can draw the line:
1. Draw a straight line through these points.
2. Extend the line in both directions, beyond the plotted points.
### Step 5: Verifying with More Points (Optional)
If you want to be more thorough, you can plot additional points:
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -5(-1) + 2 = 7 \][/tex]
So, plot the point [tex]\((-1, 7)\)[/tex].
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -5(2) + 2 = -8 \][/tex]
So, plot the point [tex]\((2, -8)\)[/tex].
### Final Graph
You should see that these points line up in a straight line, confirming that your graph is correct. When graphing, make sure to label your axes and units clearly for clarity.
Your graph should look like this:
```
Graph: y = -5x + 2
y
|
10|
9|
8|
7|
6|
5|
4|
3|
2|------o-------
1|
0|----------------------------------------- x
-1|
-2|
-3|
-4|
-5|
-6|
-7|
-8| *
```
- The point at [tex]\( (0, 2) \)[/tex] represents the y-intercept.
- The point at [tex]\( (1, -3) \)[/tex] shows the slope effect.
- Additional points should fit the same straight line.
This completes our graph of the line represented by the equation [tex]\( y = -5x + 2 \)[/tex].
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