Answer:
-6 for -2 ≤ x < 0
6 for 0 ≤ x ≤ r
Explanation:
To find the rate of change of the function g(x) over the interval -2 ≤ x ≤ r, we need to determine the equation of g(x) first. We know that the graph of g(x) is a translation 4 units down from the graph of f(x) = 6|x| - 4.
Let's start by considering the graph of f(x) = 6|x| - 4. This function represents the absolute value of x multiplied by 6 and then subtracted by 4.
To translate the graph of f(x) 4 units down, we subtract 4 from the function f(x). This results in the equation g(x) = 6|x| - 4 - 4 = 6|x| - 8.
Now, we can calculate the rate of change of g(x) over the interval -2 ≤ x ≤ r. The rate of change is determined by finding the slope of the function over the given interval.
Since the function g(x) = 6|x| - 8 is piecewise-defined (with two linear segments), we need to consider two cases: when x is negative and when x is non-negative.
Case 1: -2 ≤ x < 0 (x is negative)
In this case, the equation of g(x) simplifies to g(x) = 6(-x) - 8 = -6x - 8.
The rate of change of g(x) in this interval can be determined by finding the slope of the line -6x - 8. Since this is a linear function, the rate of change (slope) remains constant over the interval. Therefore, the rate of change of g(x) in this case is -6.
Case 2: 0 ≤ x ≤ r (x is non-negative)
In this case, the equation of g(x) simplifies to g(x) = 6(x) - 8 = 6x - 8.
Similar to the previous case, the rate of change of g(x) in this interval can be determined by finding the slope of the line 6x - 8. The rate of change (slope) remains constant over the interval, so the rate of change of g(x) in this case is 6.
Therefore, the rate of change of g(x) over the interval -2 ≤ x ≤ r is given by:
-6 for -2 ≤ x < 0
6 for 0 ≤ x ≤ r