Pre-Algebra
Functions (introductory)
Linear vs. non-linear functions
20 practice questions
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Theory + worked examples
Linear vs Non-Linear Functions
California Pre-Algebra • Standard 8.F.3 • Functions
Linear vs Non-Linear Functions is a topic in Functions in the California Common Core State Standards. It is aligned to Standard 8.F.3, which requires students to distinguish linear from non-linear functions.
A linear function has a constant rate of change and graphs as a straight line; non-linear functions curve.
Theory
A linear function has a constant rate of change and the form \(y=mx+b\); its graph is a straight line.
Non-linear functions (like \(y=x^2\)) have a changing rate and a curved graph.
A line vs a curve.
Telling them apart.
Linear form:
\[y=mx+b\]
A constant rate of change signals linear.
How to classify a function
- Check the equation for \(y=mx+b\).
- An exponent or variable in the denominator means non-linear.
- In a table, look for a constant rate of change.
- A straight-line graph is linear.
Example 1 — Identify
Is \(y=3x-2\) linear?
Solution
Yes — it fits \(y=mx+b\).
Example 2 — Non-linear
Is \(y=x^2\) linear?
Solution
No — the exponent \(2\) makes it curved.
Example 3 — From a table
A table adds \(3\) to \(y\) each time \(x\) rises by \(1\). Linear?
Solution
Constant rate of change — yes.
Example 4 — Recognize
Is \(y=\dfrac{1}{x}\) linear?
Solution
No — its graph is a curve.
Common pitfalls
\(y=x^2\) is not linear.
Constant rate of change is the linear test.
A curve is never a linear function.
Frequently asked questions
What is a linear function?
One with a constant rate of change, graphing as a line.
What makes a function non-linear?
A changing rate of change; a curved graph.
Is \(y=3x-2\) linear?
Yes.
Is \(y=x^2\) linear?
No.
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Graphing linear equations (slope-intercept form)
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