Final answer:
Linear functions are represented by the straight line equation y = mx + b, with m being the slope and b the y-intercept. These functions exhibit a constant rate of change and are graphically displayed as straight lines, whereas non-linear functions have graphs that show curves or different shapes, reflecting a variable rate of change.
Step-by-step explanation:
When we compare linear functions to non-linear functions, we look at the equations and the graphs that represent the relationships between two variables. A linear equation can be written in various forms, such as y = mx + b or y = a + bx, in which m or b is the slope and a or b is the y-intercept. The slope represents the rate of change between the dependent variable y and the independent variable x, whereas the y-intercept is the point where the line crosses the y-axis. In linear regression, these equations model the average value of y for different values of x, with the assumptions that residuals are independent, y-values are normally distributed for any value of x, and all x values have equal variance in their y values.
On a graph with perpendicular axes, a linear function is represented by a straight line which indicates a constant rate of change. The graph of this function follows the general form of y = mx + b. Non-linear functions, such as quadratic, inverse, or exponential relationships as shown in Figure 1.28, will not produce straight lines and typically represent more complex relationships in which the rate of change is not constant.