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The slope of a line is a crucial concept when graphing linear functions. It is often represented by the letter \( m \) in the linear equation \( y = mx + b \). The slope depicts how steep a line is, and it is calculated as the "rise" over the "run." This means:

• The "rise" is the vertical change between two points on a graph.
• The "run" is the horizontal change between those two points.

In the given function \( s(x) = \frac{7}{8}x + 2 \), the slope is \( \frac{7}{8} \). This tells us that for each time you move 8 units horizontally to the right, you should move 7 units vertically up. A greater slope value indicates a steeper line, while a smaller value means a flatter line. Remember, if the slope is negative, the line will decrease from left to right. This positive slope, \( \frac{7}{8} \), means our line will rise from left to right.

The y-intercept in a linear equation \( y = mx + b \) is denoted by \( b \), and it specifies where the line crosses the y-axis. In simpler terms, it's the point where the graph intersects the y-axis, which is when \( x = 0 \). For our function, \( s(x) = \frac{7}{8}x + 2 \), the y-intercept is 2. This means you start graphing by placing a point at \( (0, 2) \) on the coordinate plane.
The y-intercept is vital because it provides a fixed starting point for sketching the rest of the graph, alongside the slope. No matter the slope, every line's journey on the graph starts or crosses the y-axis at this intercept point.

The coordinate plane is a two-dimensional surface where we plot points to represent equations and solutions. It is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical). The plane makes it easy to visualize relationships between variables:

• The x-axis runs horizontally, where positive values extend to the right and negatives to the left.
• The y-axis is vertical, with positive values going upwards and negatives downwards.

For our function, we plot points based on the slope and y-intercept on this plane. The coordinate plane is fundamental in understanding geometrical interpretations of equations.

Plotting points is the foundational skill needed for graphing any function. For a linear function like \( s(x) = \frac{7}{8}x + 2 \), start by plotting the y-intercept point \( (0, 2) \). Then, use the slope \( \frac{7}{8} \) to find a second crucial point. Begin at \( (0, 2) \):

• Move 8 units to the right along the x-axis, reaching \( x = 8 \).
• From there, move 7 units up to reach \( y = 9 \).

Now place the second point at \( (8, 9) \). Connect these points with a straight line, carefully extending it in both directions.
The plotted points with the correct slope effectively represent the entire linear equation on the graph, offering a visual sense of the relationship between x and y in the equation.