The equation \( y = \frac{1}{8}x + 8 \) represents a linear function in slope-intercept form, which is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

For the given equation:
- The slope (\( m \)) is \( \frac{1}{8} \).
- The y-intercept (\( b \)) is 8.

Let's examine the key aspects of this linear function:

1. **Slope**: The slope \( \frac{1}{8} \) means that for every unit increase in \( x \), \( y \) increases by \( \frac{1}{8} \). This indicates a relatively gentle incline.

2. **Y-Intercept**: The y-intercept is 8, which means the line crosses the y-axis at the point (0, 8).

### Plotting the Equation
To plot the linear function, we can use the y-intercept and the slope:

1. **Start at the y-intercept (0, 8)**:
   - Plot the point (0, 8) on the graph.

2. **Use the slope to find another point**:
   - From (0, 8), move right 8 units (the denominator of the slope) and up 1 unit (the numerator of the slope).
   - This takes us to the point (8, 9).

3. **Draw the line**:
   - Draw a straight line through the points (0, 8) and (8, 9).

### Example Calculation
Let's calculate the value of \( y \) for a few values of \( x \):

- For \( x = 0 \):
  \[
  y = \frac{1}{8}(0) + 8 = 8
  \]

- For \( x = 8 \):
  \[
  y = \frac{1}{8}(8) + 8 = 1 + 8 = 9
  \]

- For \( x = -8 \):
  \[
  y = \frac{1}{8}(-8) + 8 = -1 + 8 = 7
  \]

### Summary
The line \( y = \frac{1}{8}x + 8 \) is a straight line with a gentle positive slope, crossing the y-axis at 8.

Would you like any further details or have any questions?

Here are 8 related questions that expand on this information:
1. What does the slope of a linear function represent?
2. How can you determine the x-intercept of the line \( y = \frac{1}{8}x + 8 \)?
3. What is the significance of the y-intercept in a linear equation?
4. How would the graph change if the slope were steeper, say \( \frac{1}{4} \)?
5. How can you determine if two lines are parallel?
6. How do you find the equation of a line given two points?
7. What is the relationship between the slopes of perpendicular lines?
8. How do changes in the slope and y-intercept affect the position and angle of a line?

**Tip:** When working with linear equations, always identify the slope and y-intercept first, as these provide crucial information for graphing and understanding the behavior of the line.

Explore the linear equation y = 1/8x + 8 and its components. Learn about slope, y-intercept, plotting the equation, and example calculations. Suitable for students in Grades 7-9.

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