Answer
To graph the function \( f(x) = \frac{1}{2}x + 7 \), plot the y-intercept at (0, 7) and another point like (2, 8). Draw a straight line through these points with a slope of \( \frac{1}{2} \).
Solution
Certainly! Let's graph the linear function:
\[
f(x) = \frac{1}{2}x + 7
\]
### 1. Understanding the Function
This is a **linear function** of the form \( f(x) = mx + b \), where:
- **Slope (\( m \))**: Determines the steepness and direction of the line.
- **Y-intercept (\( b \))**: The point where the line crosses the y-axis.
For \( f(x) = \frac{1}{2}x + 7 \):
- **Slope (\( m \))**: \( \frac{1}{2} \) (The line rises 1 unit for every 2 units it moves to the right.)
- **Y-intercept (\( b \))**: \( 7 \) (The line crosses the y-axis at (0, 7)).
### 2. Creating a Table of Values
Choose several values for \( x \) to compute corresponding \( f(x) \):
| \( x \) | \( f(x) = \frac{1}{2}x + 7 \) |
|---------|-------------------------------|
| -4 | \( \frac{1}{2}(-4) + 7 = -2 + 7 = 5 \) |
| -2 | \( \frac{1}{2}(-2) + 7 = -1 + 7 = 6 \) |
| 0 | \( \frac{1}{2}(0) + 7 = 0 + 7 = 7 \) |
| 2 | \( \frac{1}{2}(2) + 7 = 1 + 7 = 8 \) |
| 4 | \( \frac{1}{2}(4) + 7 = 2 + 7 = 9 \) |
### 3. Plotting the Points
Plot the points from the table on a coordinate plane:
- (-4, 5)
- (-2, 6)
- (0, 7)
- (2, 8)
- (4, 9)
### 4. Drawing the Line
Since it's a linear function, connect the points with a straight line. The slope \( \frac{1}{2} \) means the line rises 1 unit for every 2 units it moves to the right.
### 5. ASCII Representation of the Graph
While an exact graphical representation isn't possible in text, here's a simple ASCII sketch to illustrate the concept:
```
y
|
10|
9| * (4,9)
8| * (2,8)
7| * (0,7)
6| * (-2,6)
5| * (-4,5)
4|
3|
2|
1|
0|____________________________ x
-5 -4 -3 -2 -1 0 1 2 3 4 5
```
### 6. Description of the Graph
- **Y-Intercept**: The line crosses the y-axis at (0, 7).
- **Slope**: With a slope of \( \frac{1}{2} \), the line ascends gently.
- **Direction**: Since the slope is positive, the line moves upward from left to right.
### 7. Additional Features
- **X-Intercept**: To find where the line crosses the x-axis (\( f(x) = 0 \)):
\[
0 = \frac{1}{2}x + 7 \\
\frac{1}{2}x = -7 \\
x = -14
\]
So, the x-intercept is at (-14, 0). You can plot this point as well to extend the line further.
### 8. Visual Tools
For a more precise and visual graph, you can use graphing tools such as:
- **Desmos**: [desmos.com](https://www.desmos.com/)
- **GeoGebra**: [geogebra.org](https://www.geogebra.org/)
- **Graphing Calculators**: Physical or online calculators can plot the function.
### 9. Summary
The graph of \( f(x) = \frac{1}{2}x + 7 \) is a straight line with:
- A **positive slope** (\( \frac{1}{2} \)) indicating it rises as \( x \) increases.
- A **y-intercept** at (0, 7).
- An **x-intercept** at (-14, 0).
By plotting the key points and connecting them, you obtain the graph of the function.
Reviewed and approved by the UpStudy tutoring team
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