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The phrase "rise over run" is a simple way to understand what the slope of a line means. Imagine you're on a hill. The "rise" is how much you go up or down, and the "run" is how far you go horizontally. In mathematical terms, the rise is the change in the y-values between two points on a line, and the run is the change in the x-values.
To put it another way, rise over run indicates steepness. If the rise is positive, the line goes upwards, and if it's negative, the line goes downwards.

• A large positive rise over run means the hill is steep.
• A small rise over run means it’s gentler.
• If the rise over run is zero, the line is perfectly flat.

In this exercise, you calculated a rise (change in y-values) of \(-1\) and a run (change in x-values) of \(\frac{1}{4}\). This simplifies to a slope of \(-4\), indicating a steep, descending line.

The concept of "change in y over change in x" describes how values change as you move from one point to another on a graph. Between two points, \((x_1, y_1)\) and \((x_2, y_2)\), you calculate the change in the y-direction by doing \(y_2 - y_1\) and the change in the x-direction by doing \(x_2 - x_1\).
These changes tell us how fast things are moving vertically compared to horizontally, portraying the line's direction and steepness.

In this exercise:

• Change in y: \(-\frac{1}{4} - \frac{3}{4} = -1\)
• Change in x: \(\frac{3}{8} - \frac{1}{8} = \frac{2}{8} = \frac{1}{4}\)

This results in a slope of \(\frac{-1}{\frac{1}{4}} = -4\), suggesting that for every 1 unit moved measure horizontally (from left to right), you move 4 units down.

The slope formula is a foundational concept in algebra and geometry. It is used to determine how steep a line is, which can hint at the relationship between variables. The formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \(m\) represents the slope.
Every time you calculate a slope using two points, you apply this formula. It tells you how much y changes for each change in x, hence measuring the line's slope, effectively connecting the concepts of rise over run and change in y over change in x.

When you calculated the slope for \(\left(\frac{1}{8}, \frac{3}{4}\right)\) and \(\left(\frac{3}{8}, -\frac{1}{4}\right)\), you found \(m = \frac{-1}{\frac{1}{4}}\), which simplifies to -4, confirming again a steeply descending line. The negative slope signifies the line travels downward as it moves left to right.