91Ó°ÊÓ

The slope of a linear function signifies how steep the line is. It can be seen as the 'rise' over the 'run'. To determine the slope, you look at how much the y-value changes for a unit change in x-value. For our function, we've seen that each increase of 1 unit in x results in an increase of 4 units in y. Mathematically expressed, the slope (m) is computed as \[ m = \frac{\Delta y}{\Delta x} \] Here, \(\Delta y\) stands for the change in y, and \(\Delta x\) is the change in x. Putting the values, we get \( m = 4 \). This indicates that for every 1 unit increment in x, y increases by 4 units.

Understanding whether a function is linear or nonlinear is fundamental. A linear function forms a straight line when plotted on a graph. Its general form is \[ y = mx + c \] where \(m\) is the slope, and \(c\) is the y-intercept. A crucial characteristic of linear functions is that they have a constant rate of change. This constant rate means that the ratio of the increment in y to the increment in x stays the same, irrespective of where you are on the line. In our exercise, since the y-values increase constantly by 4 for every unit increase in x, it confirms the function is linear.

Increment analysis focuses on the changes between values of a function. By examining these increments, we can deduce important properties of the function. For the given function: \[ \begin{array}{rl} \text{When} \, x = -2, & y = -4 \ \text{When} \, x = -1, & y = 0 \ \text{When} \, x = 0, & y = 4 \ \text{When} \, x = 1, & y = 8 \ \text{When} \, x = 2, & y = 12 \ \end{array} \] We notice the changes: \[ \Delta y = \begin{cases} 4 - (-4) = 4 & \text{(from } x = -2 \text{ to } x = -1\ 4 - 0 = 4 & \text{(from } x = -1 \text{ to } x = 0) \ 8 - 4 = 4 & \text{(from } x = 0 \text{ to } x = 1) \ 12 - 8 = 4 & \text{(from } x = 1 \text{ to } x = 2)\ \end{cases} \] These consistent increments confirm the linearity of the function and help compute the slope: \( m = 4 \).

A hallmark of linear functions is the constant difference between consecutive y-values for equal increases in x-values. This constant difference ensures a straight line graphically represents the function. For our function, each move of 1 unit in x consistently results in a 4 unit increase in y: \[ \Delta y = 0 - (-4) = 4, \; 4 - 0 = 4, \; 8 - 4 = 4, \; 12 - 8 = 4 \] This predictable change, also known as the slope (\(m\)), signifies how much y changes per unit increase in x. Such constancy is essential because it defines the linearity and verifies that the slope is indeed 4, solidifying the function's linearity status.