To draw a line with a given slope and a point through which it passes, follow these steps:
### Step 1: Understand Slope-Intercept Form
A line can be described using the slope-intercept form of a linear equation:
[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 (the value of [tex]\(y\)[/tex] when [tex]\(x = 0\)[/tex]).
### Step 2: Identify Given Values
In this problem:
- The slope [tex]\(m\)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
- The line passes through the point [tex]\( (0, 4) \)[/tex]. This point provides the y-intercept [tex]\(b\)[/tex].
### Step 3: Determine the Equation of the Line
Since the point [tex]\( (0, 4) \)[/tex] lies on the line, the y-intercept [tex]\(b\)[/tex] is 4. Thus, we can write the equation of the line as:
[tex]\[ y = \frac{1}{2}x + 4 \][/tex]
### Step 4: Plot the Y-Intercept
Start by plotting the point where the line crosses the y-axis:
- The y-intercept is [tex]\(4\)[/tex], so plot the point [tex]\( (0, 4) \)[/tex].
### Step 5: Use the Slope to Find Another Point
The slope tells us how to rise and run from any point on the line to find another point on the line.
- The slope [tex]\( \frac{1}{2} \)[/tex] means that for every rise of 1 unit in the [tex]\( y \)[/tex]-direction, the line runs 2 units in the [tex]\( x \)[/tex]-direction.
From the point [tex]\( (0, 4) \)[/tex], move up 1 unit to [tex]\( y = 5 \)[/tex] and right 2 units to [tex]\( x = 2 \)[/tex]. This gives the point [tex]\( (2, 5) \)[/tex].
### Step 6: Draw the Line
1. Plot the second point [tex]\( (2, 5) \)[/tex] on the graph.
2. Draw a straight line through the points [tex]\( (0, 4) \)[/tex] and [tex]\( (2, 5) \)[/tex]. This is the line with the equation [tex]\( y = \frac{1}{2}x + 4 \)[/tex], having a slope of [tex]\( \frac{1}{2} \)[/tex] and passing through the point [tex]\( (0, 4) \)[/tex].
### Visualization
```
y
|
| .
| .
| .
| .
|------------------------------ x
0 1 2
```
Here, the points plotted [tex]\( (0, 4) \)[/tex] and [tex]\( (2, 5) \)[/tex] help you visualize and draw the best-fitting line smoothly through them.
Make sure to extend the line in both directions and draw arrows at the ends to indicate that the line extends infinitely. Label the line with its equation [tex]\( y = \frac{1}{2}x + 4 \)[/tex] for clarity.
