The lesson
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.
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.
Why this matters
Tell straight-line functions apart from curved ones.
These ideas show up on homework, quizzes, and the next unit in this grade. Getting comfortable here saves time later when the problems stack more steps together.
Common mistakes to avoid
Rushing to the answer without writing steps. Clear work helps you catch mistakes and shows your reasoning.
Re-reading the question too quickly. Underline what you are solving for before you start calculating.
Key ideas from this lesson
- Linear functions
- Nonlinear functions
Video walkthroughs
Linear and Nonlinear Functions
Decide whether a relationship can be written as a linear equation.
Watch on YouTube
Graphs of Linear Equations
Recognize straight-line graphs as linear functions.
Watch on YouTube
Finding the Slope of a Line
Compare steepness to tell linear from nonlinear patterns.
Watch on YouTube
Practice
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?