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.