Coordinate Geometry Basics: Slope as Rate of Change and Steepness

1) Slope as “Rise over Run” (Direction and Steepness)

Slope is a number that describes how a line changes as you move from left to right on a coordinate grid. It measures both:

• Direction (increasing or decreasing)
• Steepness (how quickly it rises or falls)

The basic idea is:

slope = rise / run

• Rise = vertical change (up or down)
• Run = horizontal change (right or left)

Sign interpretation (positive, negative, zero, undefined)

Rise/run examples (with signs)

• If you go up 3 and right 2, then slope = 3/2 (positive).
• If you go down 4 and right 5, then slope = -4/5 (negative).
• If you go up 0 and right 6, then slope = 0/6 = 0 (zero slope).
• If you go up 5 and right 0, then slope = 5/0 which is undefined (vertical line).

2) Slope Between Two Points: Connecting to Δy/Δx

When you are given two points, slope is computed using the changes in coordinates:

m = Δy / Δx = (y2 − y1) / (x2 − x1)

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Here, Δ (delta) means “change in.” The slope formula is just rise/run written using coordinates.

Step-by-step method

1. Label the points: (x1, y1) and (x2, y2).
2. Compute Δy = y2 − y1 (vertical change).
3. Compute Δx = x2 − x1 (horizontal change).
4. Divide: m = Δy/Δx.
5. Simplify the fraction if possible.

Example A (positive slope)

Find the slope through A(1, 2) and B(5, 10).

• Δy = 10 − 2 = 8
• Δx = 5 − 1 = 4
• m = 8/4 = 2

Interpretation: for every 1 unit right, the line goes up 2 units.

Example B (negative slope)

Find the slope through C(-2, 3) and D(4, -9).

• Δy = -9 − 3 = -12
• Δx = 4 − (-2) = 6
• m = -12/6 = -2

Interpretation: for every 1 unit right, the line goes down 2 units.

Example C (zero slope)

Find the slope through E(-1, 7) and F(6, 7).

• Δy = 7 − 7 = 0
• Δx = 6 − (-1) = 7
• m = 0/7 = 0

This is a horizontal line.

Example D (undefined slope)

Find the slope through G(3, -2) and H(3, 5).

• Δy = 5 − (-2) = 7
• Δx = 3 − 3 = 0
• m = 7/0 is undefined

This is a vertical line.

Important consistency note

You must subtract in the same order for both coordinates. For example, if you do y2 − y1, you must also do x2 − x1. If you swap the order in both places, the slope stays the same:

(y1 − y2)/(x1 − x2) gives the same value as (y2 − y1)/(x2 − x1).

3) Graph Interpretation: Comparing Slopes and Ordering by Steepness

On a grid, slope tells how steep a segment is. When comparing steepness, focus on the absolute value of slope:

steepness depends on |m|

• Larger |m| means steeper.
• Smaller |m| (closer to 0) means flatter.
• Negative slopes can be just as steep as positive ones; the sign only changes direction.

Comparing multiple segments (computed from endpoints)

Suppose four segments are drawn on the same grid with these endpoints:

• Segment 1: (0,0) to (4,2)
• Segment 2: (-1,1) to (3,9)
• Segment 3: (2,6) to (6,2)
• Segment 4: (-2,-3) to (1,3)

Compute each slope:

Order by steepness (largest |m| to smallest |m|):

• Steepest (tie): Segment 2 and Segment 4 (both |m| = 2)
• Next: Segment 3 (|m| = 1)
• Least steep: Segment 1 (|m| = 1/2)

Quick visual check using “slope triangles”

On a grid, you can compare steepness without heavy calculation by drawing a right triangle along each segment:

• Count the vertical change (rise) and horizontal change (run).
• Compute rise/run or compare ratios.
• If two segments have the same rise and run ratio, they have the same slope (they are parallel).

4) Common Issues and Targeted Practice

Issue A: Vertical lines and undefined slope

If Δx = 0, the slope is undefined because division by zero is not allowed. This happens when both points have the same x-value:

(x, y1) and (x, y2) → vertical line → undefined slope.

Tip: Before dividing, check whether x2 − x1 = 0.

Issue B: Simplifying fractions and handling negatives

Slopes are often fractions. Simplify by dividing numerator and denominator by their greatest common factor.

Example: m = 12/18 simplifies to 2/3.

Keep the negative sign in one place:

• -3/5, 3/-5, and -3/-5 (last one becomes positive 3/5)

Tip: If your slope is negative, it’s common to write it as -a/b with a positive denominator.

Issue C: Mixing up subtraction order

A frequent mistake is doing y2 − y1 but x1 − x2. That flips the sign incorrectly.

Fix: Use a consistent pattern such as:

m = (y2 − y1) / (x2 − x1)

and substitute carefully.

Targeted practice problems

1) Rise/run interpretation

• a) A line goes up 6 and right 3. What is the slope?
• b) A line goes down 5 and right 2. What is the slope?
• c) A line goes right 7 with no rise. What is the slope?
• d) A line goes up 4 with no run. What is the slope?

2) Slope between two points (compute and simplify)

• a) (2, -1) and (8, 5)
• b) (-3, 4) and (1, -8)
• c) (5, 2) and (-1, 2)
• d) (-6, 1) and (-6, 9)

3) Ordering by steepness (compute slopes, then order from steepest to least steep using |m|)

• Segment A: (0,0) to (2,6)
• Segment B: (-2,3) to (4,0)
• Segment C: (1,-1) to (5,1)
• Segment D: (3,2) to (7,10)

4) Error-checking (each student found a slope; identify whether it is correct, and if not, correct it)

• a) Points (1,2) and (4,8). Student says m = (2-8)/(4-1) = -6/3 = -2.
• b) Points (-2,5) and (-2,-1). Student says m = ( -1-5)/(-2-(-2)) = -6/0 = 0.
• c) Points (0,7) and (6,1). Student says m = (1-7)/(6-0) = -6/6 = -1.