Lesson 6: Linear Function. Slope of a Line
Lesson 6: Linear Function. Slope of a Line
Find slope as rate of change and write the equation of a line.
This lesson builds on the previous one and takes it further.
You will discover a linear function and its meaning. You will also find out how to represent a linear function in 4 different ways: equation, table of values, graph and words.
You will learn about two types of linear functions: one that passes through the origin when graphed and the other one that doesn’t.
This lesson will explain what each variable stands for in the equation
y = mx + b
When the relationship is linear, the rate of change is constant. We have already observed it in the previous lesson. The relationship between distance and time is a great example of a linear function. When graphed it produces a straight line.
The slope of a line represents the rate of change of one quantity in relation to the other. It can be calculated using any two points that sit on the line. Since each point has the coordinates (ordered pair), the slope of a line is calculated by dividing the difference between the y-coordinates by the difference between the x-coordinates.
We use letter “m” to represent the slope of a line:
m = (y2 – y1)/(x2 – x1)
Practice exercises will help you master the skills.
Key ideas: Linear Function. Slope of a Line
A linear function can be written in slope-intercept form: y = mx + b.
- m is the slope: how much y changes when x increases by 1 (the rate of change).
- b is the y-intercept: the value of y when x = 0, where the line crosses the y-axis.
- If b = 0, the line passes through the origin (direct variation).
The slope between two points (x1, y1) and (x2, y2) is m = (y2 − y1) / (x2 − x1), or “rise over run”. A positive slope rises from left to right, a negative slope falls, a horizontal line has slope 0, and a vertical line has an undefined slope.
In real situations the slope is a rate of change and has units, such as km per hour or dollars per hour.
Worked examples
Example 1
Find the slope of the line through (2, 3) and (6, 11).
- m = (11 − 3) / (6 − 2) = 8/4.
- Answer: m = 2
Example 2
Find the slope of the line through (−1, 4) and (3, −2).
- m = (−2 − 4) / (3 − (−1)) = −6/4.
- Answer: m = −3/2. The line falls from left to right.
Example 3
Find the equation of the line through (2, 3) and (6, 11).
- From Example 1, m = 2, so y = 2x + b.
- Substitute (2, 3): 3 = 2(2) + b, so b = −1.
- Answer: y = 2x − 1. Check with (6, 11): 2(6) − 1 = 11 ✓
Example 4
A plumber charges a $60 call-out fee plus $45 per hour. Write an equation and find the cost of a 3-hour job.
- The fixed fee is the y-intercept: b = 60. The rate is the slope: m = 45.
- C = 45h + 60.
- C = 45(3) + 60 = $195. The slope means the cost rises by $45 for every extra hour.
Common mistakes
- Subtracting in a different order on the top and bottom: if you start with y2 on top, start with x2 on the bottom.
- Putting the change in x on top (that gives the reciprocal of the slope).
- Confusing the y-intercept (where x = 0) with the x-intercept (where y = 0).
- Calling the slope of a vertical line 0; it is undefined.
Questions students ask
What is the slope of a horizontal line?
0, because y does not change. Its equation is y = b.
How do parallel lines relate to slope?
Parallel lines have the same slope but different y-intercepts.
Grade 9 Linear Function Quiz
Check your understanding
Try the quiz for this lesson. You’ll see right away if each answer is correct.