To graph the line given by the equation [tex]\( y - 3 = \frac{1}{4} (x + 5) \)[/tex], we will first convert the equation into slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Let's go through the steps in detail:
### Step 1: Distribute and Simplify
Start by distributing the [tex]\(\frac{1}{4}\)[/tex] on the right-hand side:
[tex]\[
y - 3 = \frac{1}{4}(x + 5)
\][/tex]
[tex]\[
y - 3 = \frac{1}{4}x + \frac{1}{4} \cdot 5
\][/tex]
[tex]\[
y - 3 = \frac{1}{4}x + \frac{5}{4}
\][/tex]
### Step 2: Isolate [tex]\( y \)[/tex]
Next, isolate [tex]\( y \)[/tex] by adding 3 to both sides of the equation:
[tex]\[
y = \frac{1}{4}x + \frac{5}{4} + 3
\][/tex]
To add 3 (which is the same as [tex]\(\frac{12}{4}\)[/tex]) to [tex]\(\frac{5}{4}\)[/tex]:
[tex]\[
y = \frac{1}{4}x + \frac{5}{4} + \frac{12}{4}
\][/tex]
[tex]\[
y = \frac{1}{4}x + \frac{17}{4}
\][/tex]
Now, the equation is in the slope-intercept form:
[tex]\[
y = \frac{1}{4}x + \frac{17}{4}
\][/tex]
### Step 3: Identify the Slope and Y-Intercept
In the equation [tex]\( y = \frac{1}{4}x + \frac{17}{4} \)[/tex]:
- The slope ([tex]\( m \)[/tex]) is [tex]\(\frac{1}{4}\)[/tex].
- The y-intercept ([tex]\( b \)[/tex]) is [tex]\(\frac{17}{4}\)[/tex], which is the point where the line crosses the y-axis ([tex]\(0, \frac{17}{4}\)[/tex]).
### Step 4: Plot the Y-Intercept
Plot the y-intercept on the graph. The point [tex]\(\left(0, \frac{17}{4}\right)\)[/tex] or [tex]\(\left(0, 4.25\right)\)[/tex] is where the line intersects the y-axis.
### Step 5: Use the Slope to Find Another Point
The slope [tex]\(\frac{1}{4}\)[/tex] means that for every 4 units increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] increases by 1 unit. Start from the y-intercept and move:
- 4 units to the right (along [tex]\(x\)[/tex]-axis),
- 1 unit up (along [tex]\(y\)[/tex]-axis).
This gives us another point on the line: [tex]\((0 + 4, 4.25 + 1) = (4, 5.25)\)[/tex].
### Step 6: Draw the Line
Using the two points [tex]\(\left(0, 4.25\right)\)[/tex] and [tex]\((4, 5.25)\)[/tex], draw a straight line through them. Extend the line across the coordinate plane.
### Step 7: Label the Axes and Provide the Graph Information
Ensure that the graph is properly labeled with [tex]\(x\)[/tex]-axis and [tex]\(y\)[/tex]-axis and add a title if necessary.
### Example Graph
Below is an example diagram to visualize the process:
```
^
| .
| /|
| / |
| / |
| . / |
| /|/ |
| / | |
|------/_|------ (0,4.25)
| / |
| / |
| / |
| / |
./ |
------------->
```
- The line starts at (0, 4.25) and follows the slope [tex]\(\frac{1}{4}\)[/tex].
### Conclusion
The graph represents the linear equation [tex]\( y - 3 = \frac{1}{4}(x + 5) \)[/tex], successfully converted to the form [tex]\( y = \frac{1}{4}x + 4.25 \)[/tex], illustrating its slope and y-intercept clearly.
