The lesson
This lesson teaches Linear & Nonlinear Functions. Read each section in order, work through every example on paper, then use the practice problems and quick check at the bottom.
Linear functions
A linear function graphs as a straight line and can be written as y = mx + b. In a table, the output changes by a constant amount each step.
When you study linear functions, slow down and write one example in your notebook without looking at the screen. That active step is what turns reading into learning.
Nonlinear functions
Anything that is not a straight line is nonlinear - for example y = x². In a table, the change between outputs is not constant.
When you study nonlinear functions, slow down and write one example in your notebook without looking at the screen. That active step is what turns reading into learning.
Why this matters
Linear & Nonlinear Functions shows up constantly in tell straight-line functions apart from curved ones. It also connects to what you will see on homework, quizzes, and the next unit in this grade.
Teachers often move fast in class. This page is here so you can pause, re-read, and practice until the idea feels familiar, not just until you have memorized a rule for one day.
Common mistakes to avoid
Rushing to the answer without writing steps. Middle-school math rewards clear work, and you catch errors earlier when steps are visible.
Mixing up similar ideas from the same topic. If two terms feel alike, make a two-column note: what is the same, what is different, and one example of each.
Key ideas from this lesson
- Linear functions
- Nonlinear functions
Video walkthrough
Linear and Nonlinear Functions
Decide whether a relationship can be written as a linear equation.
Watch on YouTubePractice
For each problem: write your work in the box, type your answer, and check it. If you are stuck, reveal the solution one step at a time. Do not skip straight to the final answer.
Exercise 1
Try it yourselfIs y = −3x + 4 a linear function?
Step-by-step solution
- 1The equation is already in slope-intercept form.
- 2The graph is a straight line.
- 3Yes, it is linear.
Exercise 2
Try it yourselfIs y = x² + 1 a linear function?
Step-by-step solution
- 1x is squared, so the rate of change is not constant.
- 2The graph is a parabola, not a line.
- 3No, it is nonlinear.
Exercise 3
Try it yourselfA table has x: 1, 2, 3, 4 and y: 5, 8, 11, 14. Is the relationship linear?
Step-by-step solution
- 1Differences in y: 8−5=3, 11−8=3, 14−11=3.
- 2Constant change of 3 for each +1 in x.
- 3Yes, the relationship is linear.
Exercise 4
Try it yourselfIs y = 5/x a linear function?
Step-by-step solution
- 1Rewriting gives xy = 5; x and y are multiplied.
- 2This is not in the form y = mx + b.
- 3No, it is nonlinear.
Exercise 5
Try it yourselfTwo functions: f(x) = 2x and g(x) = x². For x = 3, which has the greater output?
Step-by-step solution
- 1f(3) = 2(3) = 6.
- 2g(3) = 3² = 9.
- 3g(3) = 9 is greater than f(3) = 6.
Quick check
Answer all questions. Retake the quiz until you feel confident before moving on.
Linear & Nonlinear Functions
Question 1 of 4
Which equation represents a linear function?