What is the shape of the power function that controls the end-behavior?
To determine the shape of the power function that controls the end-behavior, we need to consider the degree of the function. A power function is an equation of the form f(x) = ax^n, where a is a constant and n is the degree of the function.
The end-behavior of a function refers to the behavior of the function as x approaches positive infinity or negative infinity. It helps us understand the overall shape of the function.
If the degree of the power function is even (n is an even number), then the end-behavior of the function will be the same on both sides of the y-axis. As x approaches positive infinity or negative infinity, if a > 0, the function will increase without bound, and if a < 0, it will decrease without bound. The graph of the function will resemble an "U" shape.
On the other hand, if the degree of the power function is odd (n is an odd number), then the end-behavior of the function will be different on opposite sides of the y-axis. As x approaches positive infinity, if a > 0, the function will increase without bound, and if a < 0, it will decrease without bound. As x approaches negative infinity, the behavior will be the opposite. The graph of the function will resemble an "S" shape.
Therefore, the end-behavior of a power function is determined by the degree of the function. If the degree is even, the end-behavior is the same on both sides, while if the degree is odd, the end-behavior is different on opposite sides.