Hello there. To solve this question, we'll have to remember some properties about slope of a line.
In order to find the constant rate of change of the function represented in the table below, we first have to remember why does it means that the function is a line.
Knowing the instantaneous rate of change of a function f(x) is given by its derivative, i. e.
[tex]f^{\prime}(x)[/tex]
And say this rate of change is constant, that is
[tex]f^{\prime}(x)=A[/tex]
Integrating both sides of the equation with respect to x, we have that:
[tex]\begin{gathered} \int f^{\prime}(x)dx=\int Adx \\ \\ f(x)=Ax+B \end{gathered}[/tex]
Where A and B are constants, therefore f(x) is the equation of a line.
In fact, this constant rate of change is the slope of the line, so we simply need to find it.
Given two points (x0, y0) and (x1, y1) from the equation of a line, we can find its slope m by using the following formula:
[tex]m=\frac{y_1-y_0}{x_1-x_0}[/tex]
Taking two points from the table, say (-1, 7) and (3, -9), we plug in the formula to find:
[tex]m=\frac{-9-7}{3-(-1)}=\frac{-16}{4}=-4[/tex]
Therefore the slope of this line, and constant rate of change of this function is equal to -4 and it is the answer contained in the option H.
