To graph the equation [tex]\( y = -2x + 2 \)[/tex] on the coordinate plane, follow these step-by-step instructions:
### Step 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 of the line.
- [tex]\( b \)[/tex] is the y-intercept of the line.
For the equation [tex]\( y = -2x + 2 \)[/tex]:
- The slope [tex]\( m \)[/tex] is [tex]\(-2\)[/tex].
- The y-intercept [tex]\( b \)[/tex] is [tex]\( 2 \)[/tex].
### Step 2: Plot the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. This occurs when [tex]\( x = 0 \)[/tex].
To plot the y-intercept:
- Find the point [tex]\((0, 2)\)[/tex].
- On the coordinate plane, mark a point at [tex]\((0, 2)\)[/tex].
### Step 3: Use the Slope to Find Another Point
The slope [tex]\( m = -2 \)[/tex] indicates that for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 2 units. The slope can be written as [tex]\(-2/1\)[/tex].
To find another point:
- Start from the y-intercept [tex]\((0, 2)\)[/tex].
- Move 1 unit to the right (positive direction), which corresponds to increasing [tex]\( x \)[/tex] by 1.
- From there, move 2 units down (negative direction), which corresponds to decreasing [tex]\( y \)[/tex] by 2.
This gives you a new point at [tex]\((1, 0)\)[/tex].
### Step 4: Plot the Second Point
On the coordinate plane, mark the point [tex]\((1, 0)\)[/tex].
### Step 5: Draw the Line
Using a ruler, draw a straight line passing through the points [tex]\((0, 2)\)[/tex] and [tex]\((1, 0)\)[/tex]. Extend the line in both directions, and add arrowheads to indicate that the line continues infinitely.
### Step 6: Label the Line
Label the line with its equation [tex]\( y = -2x + 2 \)[/tex].
### Verification
Optionally, you can check a few more points to ensure the accuracy of the graph. For example:
- If [tex]\( x = -1 \)[/tex], then [tex]\( y = -2(-1) + 2 = 4 \)[/tex]. Plot the point [tex]\((-1, 4)\)[/tex].
- If [tex]\( x = 2 \)[/tex], then [tex]\( y = -2(2) + 2 = -2 \)[/tex]. Plot the point [tex]\((2, -2)\)[/tex].
Here are the steps summarized in visual terms:
1. Plot the y-intercept [tex]\((0, 2)\)[/tex].
2. Use the slope to find another point, such as [tex]\((1, 0)\)[/tex].
3. Draw the line through these points and label it with the equation [tex]\( y = -2x + 2 \)[/tex].
Following these steps, you'll accurately graph the equation [tex]\( y = -2x + 2 \)[/tex] on the coordinate plane.
