Let's graph the line represented by the equation [tex]\( y = -5x + 2 \)[/tex].
### To do so, follow these steps:
1. Identify the slope and y-intercept from the equation:
- The 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.
- For the equation [tex]\( y = -5x + 2 \)[/tex]:
- The slope [tex]\( m \)[/tex] is [tex]\(-5\)[/tex].
- The y-intercept [tex]\( b \)[/tex] is [tex]\(2\)[/tex].
2. Plot the y-intercept on the graph:
- The y-intercept is the point where the line crosses the y-axis. This occurs when [tex]\( x = 0 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex]. So, the point [tex]\((0, 2)\)[/tex] is on the graph.
3. Use the slope to find another point on the graph:
- The slope [tex]\( m = -5 \)[/tex] can be interpreted as "rise over run", or in this case, a "fall" of [tex]\(5\)[/tex] units for every [tex]\(1\)[/tex] unit moved to the right.
- Starting from the y-intercept [tex]\((0, 2)\)[/tex]:
- Move [tex]\(1\)[/tex] unit to the right: [tex]\( x = 1 \)[/tex].
- Since the slope is [tex]\(-5\)[/tex], this means we move [tex]\(5\)[/tex] units down:
- New [tex]\( y \)[/tex]-value: [tex]\( 2 - 5 = -3 \)[/tex].
- The new point is [tex]\((1, -3)\)[/tex].
4. Plot this second point [tex]\((1, -3)\)[/tex] on the graph.
5. Draw the line through both points:
- Use a ruler to connect the points [tex]\((0, 2)\)[/tex] and [tex]\((1, -3)\)[/tex] with a straight line. Extend this line in both directions, and add arrowheads to indicate that it continues infinitely.
6. (Optional) Check additional points for accuracy:
- To ensure our line is correct, plug in other values of [tex]\( x \)[/tex] and solve for [tex]\( y \)[/tex]. For example, if [tex]\( x = -1 \)[/tex]:
- [tex]\( y = -5(-1) + 2 = 5 + 2 = 7 \)[/tex].
- Plot the point [tex]\((-1, 7)\)[/tex] and confirm it lies on the line.
Here is a representation of the points and the line they form:
```
y
^
10 +
9 + |
8 + |
7 + (-1, 7) |
6 + |
5 + |
4 + |
3 + |
2 + ----------------------------------------------- (+2)
1 + /
0 + /
-1 + /
-2 + /
-3 + *
-4 + |
-10 -8 -6 -4 -2 0 2 4 6 8 10 ---> x
```
In summary, the line representing the equation [tex]\( y = -5x + 2 \)[/tex] crosses the y-axis at [tex]\((0, 2)\)[/tex] and has a slope of [tex]\(-5\)[/tex], indicating it falls 5 units for every 1 unit it moves to the right.
