Introduction
Linear vs. Nonlinear Functions is an important Grade 8 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with linear vs. nonlinear functions.
What Is Linear vs. Nonlinear Functions?
Linear vs. Nonlinear Functions means choosing a model, naming what each number means, and explaining the strategy.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Linear vs. Nonlinear Functions
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Read the question carefully and identify what is being asked.
- Choose a model, equation, table, or diagram that matches the situation.
- Solve one step at a time and keep units or labels attached.
- Use the answer explanation to check that the result makes sense.
Visual Models
Visual Model 1
Question: Which table represents a LINEAR function?
| \(x\) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \(y\) | 3 | 7 | 11 | 15 |
- A. Yes, the first differences are constant
- B. No, the \(y\)-values are not all even
- C. Yes, all \(y\)-values are positive
- D. No, the \(x\)-values increase
Why it works: For a linear function, the differences between consecutive \(y\)-values are constant. Here: \(7-3=4\), \(11-7=4\), \(15-11=4\). A constant first difference indicates linearity.
Answer: Yes, the first differences are constant
Visual Model 2
Question: Which table represents a NONLINEAR function?
| \(x\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \(y\) | 1 | 2 | 4 | 8 |
- A. No, \(y\) increases
- B. Yes, the first differences are not constant
- C. Yes, the \(y\)-values are doubling
- D. No, the \(x\)-values are equally spaced
Why it works: The differences between consecutive \(y\)-values are \(2-1=1\), \(4-2=2\), \(8-4=4\). These are not constant, which indicates the function is nonlinear.
Answer: Yes, the first differences are not constant
Worked Examples
Example 1
Question: Which graph represents a LINEAR function?
- A. No, the relationship is not a function
- B. Yes, the graph is a straight line
- C. No, the slope is positive
- D. Yes, it passes through the origin
- A linear function is represented by a straight line on a coordinate plane.
- This graph shows a straight line, so the function is linear.
Answer: Yes, the graph is a straight line
Example 2
Question: Which graph represents a NONLINEAR function?
- A. Yes, the curve is parabolic
- B. No, it starts at the origin
- C. Yes, it only has positive \(y\)-values
- D. No, the points are integers
- A nonlinear function does not form a straight line.
- This graph shows a parabola (curved shape), characteristic of a quadratic or nonlinear function.
Answer: Yes, the curve is parabolic
Example 3
Question: Looking at the table below, is the function linear or nonlinear?
| \(x\) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \(y\) | 2 | 5 | 10 | 17 |
| First diff. | 3 | 5 | 7 |
- A. Linear, because \(y\) increases
- B. Nonlinear, first differences are not constant
- C. Linear, because the table has 4 points
- D. Nonlinear, because \(y > x\)
- The first differences are \(3, 5, 7\), which are not constant.
- When first differences vary, the function is nonlinear.
Answer: Nonlinear, first differences are not constant
Real-World Word Problems
Problem 1
Question: The table below shows costs. Identify whether the relationship is linear.
| Quantity | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost ($) | 12 | 24 | 36 | 48 |
- A. Yes, each item costs $12
- B. No, the cost is too high
- C. No, costs only go up
- D. Yes, because the total cost increases
Why it works: The first differences are constant: \(24-12=12\), \(36-24=12\), \(48-36=12\). This represents a linear relationship where cost increases by $12 per item.
Answer: Yes, each item costs $12
Problem 2
Question: A scientist collects data on plant height over weeks. The table shows: Is the growth linear or nonlinear?
| Week | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Height (cm) | 5 | 8 | 12 | 17 |
- A. Linear; first difference is constant at 5
- B. Nonlinear; differences are \(3, 4, 5\) (not constant)
- C. Linear; the plant always grows
- D. Nonlinear; cannot predict height after week 3
Why it works: First differences: \(8-5=3\), \(12-8=4\), \(17-12=5\). Non-constant differences indicate nonlinear growth.
Answer: Nonlinear; differences are \(3, 4, 5\) (not constant)
Common Mistakes
- Rushing before identifying what the numbers represent.
- Choosing an operation that does not match the situation.
- Dropping labels, units, or context from the answer.
- Skipping the estimate or reasonableness check.
Strategy Tips
- Underline the question being asked.
- Use a model before jumping to computation.
- Write an equation that matches the story or picture.
- Explain the final answer in a sentence.
Practice Questions
Question 1
Which equation represents a NONLINEAR function?
- A. \(y=2x+3\)
- B. \(y=-x+5\)
- C. \(y=x^2+1\)
- D. \(y=\frac{1}{2}x-7\)
Question 2
Which of the following equations represents a LINEAR function?
- A. \(y=x^3-2\)
- B. \(y=\sqrt{x}\)
- C. \(y=\frac{1}{x}+1\)
- D. \(y=5-4x\)
Question 3
Which equation is LINEAR?
- A. \(y=3x^2\)
- B. \(y=\frac{2}{x}\)
- C. \(y=6x-9\)
- D. \(y=\sqrt{2x}\)
Question 4
Which equation is NONLINEAR?
- A. \(y=-7x\)
- B. \(y=4-x\)
- C. \(y=x^4-3\)
- D. \(y=\frac{1}{3}x+2\)
Question 5
Which table shows a LINEAR function?
| \(x\) | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| \(y\) | 5 | 11 | 17 | 23 |
- A. Yes; the \(y\)-values increase by 6 each step
- B. No; \(x\) increases by 2 each time
- C. No; \(y\) is greater than \(x\)
- D. No; the equation is \(y=3x-1\)
Question 6
The function \(y=\sqrt{x-1}\) is LINEAR or NONLINEAR?
- A. Linear, because it has a constant slope
- B. Nonlinear, because \(x \geq 1\)
- C. Linear, because \(x\) appears once
- D. Nonlinear, because it contains a square root
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(y=x^2+1\)
A linear function has the form \(y=mx+b\) (no powers of \(x\) other than 1). \(y=x^2+1\) has a squared variable, making it nonlinear.
Question 2
Answer: \(y=5-4x\)
The equation \(y=5-4x\) is in the form \(y=mx+b\) with \(m=-4\) and \(b=5\), so it is linear. The other functions contain exponents (3), square roots, or reciprocals, making them nonlinear.
Question 3
Answer: \(y=6x-9\)
Only \(y=6x-9\) is in the form \(y=mx+b\). Options A, B, and D involve exponents, reciprocals, or square roots, making them nonlinear.
Question 4
Answer: \(y=x^4-3\)
\(y=x^4-3\) contains an exponent of 4, making it nonlinear. Options A, B, and D all have the form \(y=mx+b\), so they are linear.
Question 5
Answer: Yes; the \(y\)-values increase by 6 each step
The \(y\)-values are \(5, 11, 17, 23\). The differences are \(11-5=6\), \(17-11=6\), and \(23-17=6\). Because the difference is constant for equal steps in \(x\), the function is linear.
Question 6
Answer: Nonlinear, because it contains a square root
The presence of a square root makes this function nonlinear. Linear functions can only have \(x\) to the first power and no radicals.
Connection to Standards
This lesson supports Grade 8 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Linear vs. Nonlinear Functions becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Understand the model before choosing the operation.
