8.4 Slope & Coordinate Basics

The Coordinate Plane

The coordinate plane is a flat surface formed by two perpendicular number lines. The horizontal line is the x-axis, and the vertical line is the y-axis. They cross at a point called the origin, which has coordinates (0, 0). Every point is described by an ordered pair (x, y): the first number (the x-coordinate) tells you how far to move left or right from the origin, and the second number (the y-coordinate) tells you how far to move up or down.

To plot (3, 4): start at the origin, move 3 right, then 4 up. To plot (−2, 5): move 2 left, then 5 up. To plot (−4, −1): move 4 left, then 1 down.

The axes divide the plane into four quadrants: Quadrant I (upper right, x > 0, y > 0), Quadrant II (upper left, x < 0, y > 0), Quadrant III (lower left, x < 0, y < 0), and Quadrant IV (lower right, x > 0, y < 0). Points on the axes are not in any quadrant.

Common trap: The order of the coordinates matters. The point (3, 5) is different from (5, 3). The first coordinate is always horizontal (x), and the second is always vertical (y). Swapping them plots a different point.

Slope: Rise over Run

The slope of a line measures how steep it is and which direction it tilts. It is defined as the ratio of the vertical change to the horizontal change between two points on the line. The formula, using any two points (x₁, y₁) and (x₂, y₂) on the line, is:

m = (y₂ − y₁) / (x₂ − x₁)

The vertical change is called the rise, and the horizontal change is called the run, so slope is rise over run. The letter m is the standard symbol for slope.

Example 1. Find the slope of the line passing through (1, 2) and (4, 8).

Label the points: (x₁, y₁) = (1, 2) and (x₂, y₂) = (4, 8).

Compute the rise: y₂ − y₁ = 8 − 2 = 6.

Compute the run: x₂ − x₁ = 4 − 1 = 3.

Divide: m = 6 / 3 = 2.

The slope is 2, meaning the line rises 2 units for every 1 unit it moves to the right.

Example 2. Find the slope through (−2, 5) and (3, −1).

Rise: y₂ − y₁ = −1 − 5 = −6.

Run: x₂ − x₁ = 3 − (−2) = 5.

Slope: m = −6 / 5. The slope is negative, so the line falls from left to right.

Common trap (rise over run vs run over rise): Slope is vertical change divided by horizontal change. A frequent error is to flip the order and compute (x₂ − x₁) / (y₂ − y₁). Always subtract y-values on top and x-values on the bottom. A useful check: if the line is going up to the right, the slope should be positive; if your calculation gives a negative number for a clearly rising line, you flipped the order.

The Four Types of Slope

1. Positive slope: The line rises from left to right. The y-values increase as x increases. Example: m = 2.
2. Negative slope: The line falls from left to right. The y-values decrease as x increases. Example: m = −3.
3. Zero slope: The line is horizontal. The y-values do not change at all, so y₂ − y₁ = 0, and m = 0 / (run) = 0. Every point on a horizontal line has the same y-coordinate.
4. Undefined slope: The line is vertical. The x-values do not change, so x₂ − x₁ = 0, and division by zero is undefined. Every point on a vertical line has the same x-coordinate.

Common trap (vertical line slope): A vertical line has an undefined slope, not a slope of zero. A slope of zero is a horizontal line. The confusion comes from dividing by zero; since division by zero is not allowed, the slope is undefined, not 0. Remember: horizontal = slope 0, vertical = undefined.

Slope-Intercept Form: y = mx + b

A linear equation written as y = mx + b is in slope-intercept form. In this form:

• m is the slope of the line.
• b is the y-intercept, the y-coordinate of the point where the line crosses the y-axis. (Because x = 0 on the y-axis, the y-intercept point is (0, b).)

For example, in y = 2x + 3, the slope is 2 and the y-intercept is 3 (the line crosses the y-axis at (0, 3)). In y = −x − 4, the slope is −1 and the y-intercept is −4.

Graphing a Line from Slope-Intercept Form

To graph a line given y = mx + b:

1. Plot the y-intercept (0, b) on the y-axis.
2. From that point, use the slope m = rise / run to find a second point. A positive rise moves up; a negative rise moves down. A positive run moves right; a negative run moves left.
3. Draw a straight line through the two points.

Example 3. Graph y = (1/2)x + 3.

Step 1: Plot the y-intercept (0, 3). Step 2: The slope is 1/2, so from (0, 3) move up 1 and right 2 to reach (2, 4). Step 3: Draw the line through (0, 3) and (2, 4).

Example 4. Graph y = −2x + 1.

Step 1: Plot the y-intercept (0, 1). Step 2: The slope is −2, so from (0, 1) move down 2 and right 1 to reach (1, −1). Step 3: Draw the line through (0, 1) and (1, −1).

Finding Slope from Two Points

The slope formula works with any two points on a line. The order of the points does not matter as long as you are consistent—whatever point you label as (x₁, y₁) in the numerator must also be (x₁, y₁) in the denominator.

Example 5. Find the slope of the line through (2, −3) and (5, 6).

Rise: 6 − (−3) = 9. Run: 5 − 2 = 3. Slope: m = 9 / 3 = 3.

Example 6. Find the slope through (−1, 4) and (3, 4).

Rise: 4 − 4 = 0. Run: 3 − (−1) = 4. Slope: m = 0 / 4 = 0. The line is horizontal.

Example 7. Find the slope through (2, −1) and (2, 5).

Rise: 5 − (−1) = 6. Run: 2 − 2 = 0. Slope: m = 6 / 0, which is undefined. The line is vertical.

Common Traps Recap

• Swapping x and y coordinates. The ordered pair (3, 5) is not the same as (5, 3).
• Rise over run vs run over rise. Slope is vertical change over horizontal change. Put the y-difference on top.
• Vertical line slope is undefined, not zero. Zero slope is a horizontal line; undefined slope is a vertical line.
• Sign of the slope. If the line falls from left to right, the slope is negative. A positive calculation for a falling line means you subtracted in the wrong order.