a. To graph the function f(x) = 2|x| over the domain -4 ≤ x ≤ 4, we consider the absolute value function |x|. The absolute value of a number gives us its distance from zero on the number line. Since we are taking the absolute value of x, the function will always be positive or zero.
For x < 0, the expression inside the absolute value brackets, -x, will be positive. Therefore, f(x) = 2(-x) = -2x.
For x ≥ 0, the expression inside the absolute value brackets, x, will be positive. Therefore, f(x) = 2x.
We can plot the points for both cases and obtain the graph as follows:
For x < 0:
When x = -4, f(x) = 2(-4) = -8.
When x = -3, f(x) = 2(-3) = -6.
When x = -2, f(x) = 2(-2) = -4.
When x = -1, f(x) = 2(-1) = -2.
For x ≥ 0:
When x = 0, f(x) = 2(0) = 0.
When x = 1, f(x) = 2(1) = 2.
When x = 2, f(x) = 2(2) = 4.
When x = 3, f(x) = 2(3) = 6.
When x = 4, f(x) = 2(4) = 8.
Plotting these points, we get a V-shaped graph with the vertex at (0, 0) and the arms extending upwards and downwards.
b. The rate of change over the interval 0 ≤ x ≤ 4 can be determined by calculating the slope of the line connecting two points on the graph. Taking any two points on the interval, for example, (0, 0) and (4, 8), we can calculate the slope as (change in y) / (change in x):
Slope = (8 - 0) / (4 - 0) = 8 / 4 = 2.
Therefore, the rate of change over the interval 0 ≤ x ≤ 4 is 2.
c. The rate of change over this interval, which is 2, is constant. It means that for every unit increase in x, the value of f(x) increases by 2. This is consistent with the linear relationship between x and f(x) within this interval. The graph of the function f(x) = 2|x| has a constant slope because the absolute value function changes linearly, resulting in a consistent rate of change for the function f(x).