Slope measures how steeply a line rises or falls. It is defined as the ratio of vertical change (rise) to horizontal change (run) between any two points on the line: m = (y₂ − y₁)/(x₂ − x₁). A positive slope means the line rises left to right; a negative slope means it falls; slope = 0 means horizontal; an undefined slope means vertical (division by zero).
Because a line has constant slope everywhere, you only need two points to calculate it.
The slope-intercept form is the most common way to write a linear equation. Here is the slope and is the y-intercept (the y-value where the line crosses the y-axis). This form makes graphing straightforward: start at and use the slope to find additional points.
Point-slope form is useful when you know a point and the slope but not the y-intercept. Two lines with the same slope are parallel; perpendicular lines have slopes that are negative reciprocals of each other ().
Understanding Slope and Linear Equations
A linear equation describes a constant rate of change. This is the main idea behind the graph, not just a rule for plotting points. If a taxi charge rises by the same amount for every mile traveled, a line can model the cost.
If a tank fills at the same number of liters each minute, a line can model the water level. The slope then has units.
When the horizontal axis is time in hours and the vertical axis is distance in kilometers, the slope means kilometers per hour. Checking units helps make an answer meaningful and can reveal a mistake.
Careful subtraction matters when finding a slope from a table or graph. Keep the coordinates of each point together. Find the vertical change using one point order, then find the horizontal change using that same order.
Reversing both orders gives the same result. Reversing only one order changes the sign and produces a wrong answer. Negative coordinates need extra care because subtracting a negative value increases the result.
It is often helpful to write each change in words before calculating. A downward change of six units is negative six, while a movement left of three units is negative three.
Different equation forms are useful for different jobs. Slope-intercept form is efficient when the starting value is known. In a real situation, the intercept often represents an amount present at zero time, such as a basic fee before any travel occurs.
Point-slope form is useful when a line must pass through a known measurement. To change point-slope form into slope-intercept form, distribute the slope through the parentheses, then isolate the vertical variable. Standard form is often helpful when finding where a line crosses each axis.
Set the horizontal variable to zero to find the vertical intercept. Set the vertical variable to zero to find the horizontal intercept. Vertical lines need separate treatment because they cannot be described by a single output for every input.
Graphs can be misleading when the axis scales are different. A line may look steep on paper because the vertical scale is stretched, even if its numerical slope is small. Read the numbered scale before counting rise and run.
Real data may not form a perfect line either. A scatter plot of student heights and shoe sizes, or temperatures measured through a day, can have variation. A line of best fit summarizes the general trend, but it does not make every point exact.
When using a linear model, pay attention to whether a constant rate makes sense over the interval being studied. A car can travel at a roughly steady speed for a short period, but traffic and stops make the relationship less reliable over a long trip.
Key Facts
- Slope:
- Slope-intercept form: (, )
- Point-slope form:
- Standard form: (useful for and intercepts)
- Parallel lines: same slope (); perpendicular lines: negative reciprocal slopes ()
- Horizontal line: slope = , equation ; vertical line: undefined slope, equation
Vocabulary
- Slope
- The ratio of vertical change to horizontal change between two points on a line; measures steepness and direction.
- Y-intercept
- The point where a line crosses the y-axis (where ); the value in .
- Slope-intercept form
- The equation form , where is the slope and is the y-intercept; directly shows the rate of change and starting value.
- Linear equation
- An equation whose graph is a straight line; all variables are to the first power and no variable is multiplied by another.
- Rate of change
- How much one quantity changes per unit change in another; for a linear function, this is constant and equal to the slope.
Common Mistakes to Avoid
- Computing slope as instead of . Slope = (vertical over horizontal). Reversing this gives the reciprocal, not the slope.
- Confusing the y-intercept with the x-intercept. The y-intercept ( in ) is where the line crosses the y-axis (set ). The x-intercept is where it crosses the x-axis (set and solve for ).
- Assuming any two points give a different slope for the same line. Slope is constant everywhere on a straight line - it doesn't matter which two points you choose.
- Misapplying the perpendicular slope rule. Perpendicular slopes are negative reciprocals: if m = 2/3, the perpendicular slope is −3/2. Both the sign change and the flip are required.
Practice Questions
- 1 Find the slope and y-intercept of the line passing through (−2, 5) and (4, −1). Write the equation in slope-intercept form.
- 2 Line A has equation y = (3/4)x − 2. Write the equation of a line perpendicular to A passing through the point (0, 5).
- 3 A car travels at a constant speed. After 2 hours it has gone 120 km; after 5 hours it has gone 300 km. What is the slope of the distance-time graph and what does it represent physically?