Table of Contents
Understanding Slope
In the broader topic of linear functions, slope is the number that tells you how steep a line is and in which direction it tilts. This chapter focuses on what slope means, how to compute it, and how to interpret it in different contexts.
The Idea of Slope: “Rise over Run”
A straight line on a coordinate plane has a constant steepness. That steepness is its slope.
Imagine walking along a straight ramp from left to right:
- The vertical change (how much you go up or down) is called the rise.
- The horizontal change (how far you go forward) is called the run.
The slope is defined as:
$$
\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\text{vertical change}}{\text{horizontal change}}.
$$
If you move from one point on the line to another:
- If you go up as you go to the right, the slope is positive.
- If you go down as you go to the right, the slope is negative.
- If you neither go up nor down (a flat line), the slope is zero.
- If the line is perfectly vertical, its slope is undefined (you would be dividing by zero in the formula).
Slope Between Two Points
A line can be described by any two distinct points on it. Suppose you have two points:
- $P_1 = (x_1, y_1)$
- $P_2 = (x_2, y_2)$
The slope $m$ of the line through these points is:
$$
m = \frac{y_2 - y_1}{x_2 - x_1}.
$$
Here:
- $y_2 - y_1$ is the change in $y$ (vertical change).
- $x_2 - x_1$ is the change in $x$ (horizontal change).
You may also see this written as:
$$
m = \frac{\Delta y}{\Delta x},
$$
where $\Delta$ (the Greek letter “delta”) means “change in”.
Important details about the formula
- The order must be consistent. If you do $y_2 - y_1$ on top, you must do $x_2 - x_1$ on the bottom, with the same order of points.
- It does not matter which point you call $(x_1, y_1)$ or $(x_2, y_2)$, as long as you are consistent in the subtraction.
For example, between $(2, 3)$ and $(5, 9)$:
$$
m = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2.
$$
If you swap the roles of the points:
$$
m = \frac{3 - 9}{2 - 5} = \frac{-6}{-3} = 2,
$$
which is the same result.
Interpreting the Sign and Size of Slope
Slope tells you both the direction and the steepness of a line.
Direction (sign of the slope)
- Positive slope ($m > 0$):
- The line goes up as you move from left to right.
- Example: $m = 2$, $m = \frac{1}{3}$.
- Negative slope ($m < 0$):
- The line goes down as you move from left to right.
- Example: $m = -1$, $m = -\frac{5}{2}$.
- Zero slope ($m = 0$):
- The line is horizontal (flat).
- There is no vertical change when $x$ changes.
- Undefined slope:
- The line is vertical.
- The $x$-coordinate does not change, so $x_2 - x_1 = 0$, and
$$
\frac{y_2 - y_1}{x_2 - x_1}
$$
would require dividing by zero, which is not allowed.
Steepness (magnitude of the slope)
The size (absolute value) of the slope tells you how steep the line is.
- A larger absolute value means a steeper line.
- $m = 5$ is steeper than $m = 2$.
- $m = -4$ is steeper (in magnitude) than $m = -1$.
- A smaller absolute value (between 0 and 1) means the line is less steep.
- $m = \frac{1}{2}$ is less steep than $m = 3$.
- $m = -\frac{1}{4}$ is less steep than $m = -2$.
Slope as a Rate of Change
Slope is not just a geometric idea; it has a natural meaning in many real-world situations: it represents a rate of change.
If a linear function describes how one quantity depends on another, the slope tells you:
“How much does the output change when the input increases by 1?”
In more formal terms, if $y$ depends on $x$ through a linear relationship, the slope $m$ answers:
$$
\text{For each increase of 1 in } x, \text{ how much does } y \text{ change?}
$$
Typical interpretations
- In a distance–time graph (distance on the vertical axis, time on the horizontal axis), the slope is the speed.
- In a cost–quantity graph (cost vs. number of items), the slope can be the cost per item.
- In a temperature–time graph (temperature vs. time), the slope might be the temperature change per unit of time.
The units of slope come from:
$$
\text{slope} = \frac{\text{change in dependent variable}}{\text{change in independent variable}}.
$$
For example, if $y$ is distance in kilometers and $x$ is time in hours, then:
$$
\text{slope units} = \frac{\text{kilometers}}{\text{hours}} = \text{km/h},
$$
which is a speed.
Slope from a Graph
When you are given a graph of a line, you can find its slope visually.
- Choose two points on the line whose coordinates you know or can read accurately from the grid.
- Starting from the first point, count the vertical change (rise) to reach directly above or below the second point.
- Then count the horizontal change (run) to reach the second point.
- Use:
$$
m = \frac{\text{rise}}{\text{run}}.
$$
The rise may be positive (going up) or negative (going down). The run is positive when you move to the right.
For example, if from one point to another you go up 3 units and right 2 units, then:
$$
m = \frac{3}{2}.
$$
If from one point to another you go down 4 units and right 1 unit, then:
$$
m = \frac{-4}{1} = -4.
$$
Special Cases: Horizontal and Vertical Lines
Two particularly important special cases are:
Horizontal lines
A horizontal line has the same $y$-value for every point on it.
- Any two points look like $(x_1, c)$ and $(x_2, c)$, where $c$ is constant.
- Then:
$$
m = \frac{c - c}{x_2 - x_1} = \frac{0}{x_2 - x_1} = 0.
$$ - So horizontal lines have slope $0$.
Vertical lines
A vertical line has the same $x$-value for every point on it.
- Any two points look like $(k, y_1)$ and $(k, y_2)$, where $k$ is constant.
- Then:
$$
m = \frac{y_2 - y_1}{k - k} = \frac{y_2 - y_1}{0},
$$
which is undefined (division by zero). - So vertical lines have undefined slope.
These cases are important to recognize quickly from graphs or equations.
Slope in Different Forms of Linear Equations
The detailed study of equation forms belongs to other chapters, but it is useful here to know how slope appears in the most common form.
In the slope–intercept form of a line:
$$
y = mx + b,
$$
the coefficient $m$ is the slope.
- Changing $m$ changes the tilt of the line.
- A larger $m$ (in absolute value) means the line is steeper.
- A positive $m$ makes the line rise to the right; a negative $m$ makes it fall to the right.
Recognizing $m$ as the slope makes it easy to:
- Compare how steep different lines are.
- Understand how quickly one quantity is changing with respect to another in a real-world model.
Summary of Key Points
- Slope measures the steepness and direction of a line.
- Formula between two points $(x_1, y_1)$ and $(x_2, y_2)$:
$$
m = \frac{y_2 - y_1}{x_2 - x_1}.
$$ - Positive slope: line rises to the right.
Negative slope: line falls to the right. - Zero slope: horizontal line.
Undefined slope: vertical line. - Slope is a rate of change: change in output per unit change in input.
- In $y = mx + b$, the number $m$ is the slope.