End Behavior of Functions

Front

What is end behavior of a function?

Back

End behavior of a function describes the behavior of the function as the input values (x) approach positive or negative infinity.

Front

How do you determine the end behavior of a polynomial function?

Back

To determine the end behavior of a polynomial function, analyze the leading term's degree and coefficient. If the degree is even and the leading coefficient is positive, both ends go to positive infinity. If the degree is even and the leading coefficient is negative, both ends go to negative infinity. If the degree is odd and the leading coefficient is positive, the left end goes to negative infinity and the right end goes to positive infinity. If the degree is odd and the leading coefficient is negative, the left end goes to positive infinity and the right end goes to negative infinity.

Front

What does it mean if a function approaches negative infinity as x approaches positive infinity?

Back

It means that as the input values increase without bound, the output values of the function decrease without bound.

Front

What is the significance of the leading term in determining end behavior?

Back

The leading term of a polynomial function has the greatest impact on the function's end behavior as it dominates the function's value for very large or very small x values.

Front

For the function f(x) = 1 + x^7 - x^5, what is the end behavior as x approaches negative infinity?

Back

As x approaches negative infinity, f(x) approaches negative infinity.

Front

For the function f(x) = 1 + x^4 - x^5 + 2x^2, what is the end behavior as x approaches positive infinity?

Back

As x approaches positive infinity, f(x) approaches negative infinity.

Front

What does it indicate if a function's graph rises to positive infinity on both ends?

Back

It indicates that the leading term of the function is even and has a positive coefficient.

Front

How can you visually determine the end behavior of a function from its graph?

Back

You can visually determine the end behavior by observing the direction in which the graph heads as x approaches positive and negative infinity.

Front

What is the end behavior of the function f(x) = x^3 as x approaches negative infinity?

Back

As x approaches negative infinity, f(x) approaches negative infinity.

Front

What is the end behavior of the function f(x) = -2x^2 as x approaches positive infinity?

Back

As x approaches positive infinity, f(x) approaches negative infinity.