Final answer:
The end behavior of the function f(x)=2|x+2|-6 is that as x approaches either positive or negative infinity, the function approaches infinity. The graph of f(x) will be V-shaped, with the lowest point at x = -2, shifted downward by 6 units.
Step-by-step explanation:
To determine the end behavior of the function f(x)=2|x+2|-6, we need to consider the behavior of the function as x approaches negative and positive infinity. Since the term |x+2| represents the absolute value, it will be positive regardless of whether x is positive or negative.
As x approaches positive infinity (x → ∞), the term |x+2| will also approach infinity, and thus f(x) will approach ∞ as well. As x approaches negative infinity (x → -∞), the term |x+2| still approaches infinity because the absolute value of a large negative number is a large positive number. Therefore, in both cases, f(x) increases without bound, minus the constant (-6). The '-6' merely shifts the function down on the y-axis but does not affect the overall upward trend of the function as x approaches either infinity.
Ultimately, the end behavior of f(x) is such that both as x → ∞ and x → -∞, the function f(x) will approach infinity. The graph of f(x) will be V-shaped, with the lowest point at x = -2 and shifted downward by 6 units.