Final answer:
To determine the value of a so that the table data represents a linear function with a rate of change of -8, one needs to set the slope of the function to -8 and solve for a using two points from the table.
Step-by-step explanation:
The student is asking about linear functions, specifically how to determine the value of a in a table of data so that the function has a rate of change of -8. This problem involves understanding what constitutes a linear function and how the slope, or rate of change, of the function is determined and utilized.
To address this, we need to consider the general form of a linear function, which is y = mx + b, where m is the slope, x is the independent variable, and b is the y-intercept. When a linear function has a rate of change of -8, the slope (m) is -8. This means that for every one unit increase in x, the value of y decreases by 8 units.
In a table representing a linear function, the differences between consecutive y-values should be consistent when x values increase by a constant amount. Therefore, the value of a should be chosen such that when you calculate the difference between the y-values where a is used and the preceding y-value, the result must be -8 times the difference in the x-values.
An example of how to calculate this would be to take two points from the given table, (x1, y1) and (x2, y2), where x2 > x1 and y2 includes the variable a. Apply the formula (y2 - y1) / (x2 - x1) to find the slope. Set this equal to -8 and solve for a. This will give you the necessary value of a so that the rate of change of the function represented by the table is -8.