To describe the rate of change of the function \( f(x) = -6x - 7 \), we first need to understand that the rate of change in the context of a function, particularly a linear function, is another term for its slope.
Now, let's recall what the slope of a linear function means. A linear function is of the form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. The slope \( m \) determines how steep the line is, and it represents the change in the y-value (the output of the function) for a unit change in the x-value (the input of the function).
Looking at the given function \( f(x) = -6x - 7 \), we can see it is in the same form as \( y = mx + b \), with the slope \( m = -6 \) and the y-intercept \( b = -7 \).
The slope being -6 means that for every one unit increase in \( x \), the function's value decreases by 6 units. This is a constant rate of change because it does not matter what the value of \( x \) is; the function will always decrease by 6 for every one unit increase in \( x \).
Now, let's examine the given statements:
A. The function has a varying rate of change when \( x < 7 \).
- This statement is incorrect because the slope is constant at -6, regardless of the value of \( x \).
B. The function has a constant rate of change, decreasing for all \( x \) at a rate of 6.
- This statement is correct; it accurately describes the behavior of a linear function with a slope of -6.
C. The function has a varying rate of change when \( x < 6 \).
- This statement is also incorrect because, once again, the rate of change (slope) is constant for all values of \( x \).
D. The function has a constant rate of change, decreasing for all \( x \) at a rate of 7.
- This statement is incorrect; the coefficient of \( x \) in the function, which represents the slope, is -6, not -7.
Based on our analysis, the correct answer is:
B. The function has a constant rate of change, decreasing for all \( x \) at a rate of 6.