Final answer:
The end behavior of the function f(x) = -|x - 2| + 3 is a downward trend as x approaches positive infinity and an upward trend as x approaches negative infinity.
Step-by-step explanation:
The end behavior of a function refers to what happens to the function as x approaches positive infinity and negative infinity. To determine the end behavior of the function f(x) = -|x - 2| + 3, we need to look at the behavior of the function when x is very large positive and negative values.
- For large positive values of x, the expression |x - 2| will simplify to x - 2 since x - 2 is positive.
- So, the function becomes -x + 2 + 3 = -x + 5.
- As x approaches positive infinity, the function will become more and more negative, resulting in a downward trend.
Similarly, for large negative values of x, the expression |x - 2| will simplify to -(x - 2) = -x + 2 since x - 2 is negative.
- So, the function becomes -(-x + 2) + 3 = x - 2 + 3 = x + 1.
- As x approaches negative infinity, the function will become more and more positive, resulting in an upward trend.
- Therefore, the end behavior of the function f(x) = -|x - 2| + 3 is a downward trend as x approaches positive infinity and an upward trend as x approaches negative infinity.