Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What characterizes linear functions?

1. They have a variable slope

2. The slope is constant

3. They are represented as curves

4. They include exponential components

The correct answer is: The slope is constant

Linear functions are characterized by having a constant slope. This means that for any two points on a linear function, the change in the y-value divided by the change in the x-value is the same, regardless of which two points are chosen. This constant rate of change is what distinguishes linear functions from other types of functions, such as quadratic or exponential functions, which have variable slopes and do not maintain a consistent rate of change throughout. In graphical terms, the constant slope results in the graph of a linear function being a straight line, as opposed to curves or other shapes seen in non-linear functions. The equation of a linear function typically takes the form \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept. This structure reinforces the idea that the relationship between x and y is linear, highlighting the linearity of the function. The other options present characteristics that do not apply to linear functions, such as a variable slope, representation as curves, or the inclusion of exponential components, which inherently describe non-linear behaviors.