Describe the differences between a linear function and an exponential function.
my answer is Linear functions have a constant rate of change while an exponential function has a constant finite ratio.
sounds good.
You might also note that for an exponential function y = f(x), log(y) is linear.
To understand the differences between a linear function and an exponential function, let's break down each function and look at their key characteristics.
1. Linear Function:
- A linear function is a mathematical equation of the form y = mx + b, where x and y represent variables, m represents the slope (or rate of change), and b represents the y-intercept (the point where the line crosses the y-axis).
- The key characteristic of a linear function is that its rate of change remains constant. This means that for every unit increase in x, the corresponding change in y will be the same.
- Linear functions graphically appear as straight lines on a coordinate plane.
2. Exponential Function:
- An exponential function is a mathematical equation of the form y = ab^x, where x and y represent variables, a represents the initial value, and b represents the base.
- The key characteristic of an exponential function is that it has a constant finite ratio. This means that there is a consistent increase or decrease by a constant factor for each unit increase in x.
- Exponential functions graphically appear as curves that either increase or decrease exponentially on a coordinate plane.
While your answer is close, it is important to note that the rate of change for a linear function is constant, whereas an exponential function involves a constant finite ratio.