A function is linear if it satisfies two main criteria:

1. It can be written in the form $f(x) = mx + b$, where $m$ and $b$ are constants:

   • Here, $m$ represents the slope, indicating the rate of change of the function, and $b$ is the y-intercept, showing where the function crosses the y-axis.
   • This form ensures that the graph of the function is a straight line.

2. It has a constant rate of change (slope):

   • In a linear function, the difference in $f(x)$ values for equal differences in $x$ values remains constant.
   • This can be tested by calculating the slope between any two points on the function. If the slope remains the same for all pairs of points, the function is linear.

Properties of Linear Functions

• The graph is always a straight line.
• The degree (highest power of $x$) in a linear equation is 1, meaning no exponents on $x$ are greater than 1.
• Linear functions do not have curves, sharp turns, or bends in their graph.

Would you like to see a specific example of testing whether a function is linear?

Related Questions:

1. What distinguishes a linear function from a nonlinear one?
2. How do you calculate the slope of a linear function?
3. Can a linear function have more than one variable?
4. What is the role of the y-intercept in linear functions?
5. How does a linear function differ from an affine function?

Tip:

For any given function, if you see terms like $x^2$ or $x/y$, the function is nonlinear.