End Behavior of a Function
| Degree | Leading Coefficient | End behavior of the function | Graph of the function |
| Even | Positive | f ( x) → + ∞, as x → − ∞ f ( x) → + ∞, a ... | Example: f ( x) = x 2 |
| Even | Negative | f ( x) → − ∞, as x → − ∞ f ( x) → − ∞, a ... | Example: f ( x) = − x 2 |
| Odd | Positive | f ( x) → − ∞, as x → − ∞ f ( x) → + ∞, a ... | Example: f ( x) = x 3 |
| Odd | Negative | f ( x) → + ∞, as x → − ∞ f ( x) → − ∞, a ... | Example: f ( x) = − x 3 |
Full Answer
How do you determine end behavior?
- If n < m, the horizontal asymptote is y = 0.
- If n = m, the horizontal asymptote is y = a/b.
- If n > m, there is no horizontal asymptote.
How would you describe the end behavior?
Determining end behavior algebraically
- Investigation: End behavior of monomials. Monomial functions are polynomials of the form , where is a real number and is a nonnegative integer.
- Concluding the investigation. Notice how the degree of the monomial and the leading coefficient affect the end behavior. ...
- End behavior of polynomials. We now know how to find the end behavior of monomials. ...
How to find the end behavior?
How To Find End Behavior Asymptote? Step 1: Look at the degrees of the numerator and denominator. If the degree of the denominator is larger than the degree of the numerator, there is a horizontal asymptote of y=0, which is the end behavior of the function.
How to determine the end behavior of a rational function?
How to Determine the End Behavior of a Rational Function
- Determining the End Behavior of a Rational Function. Step 1: Look at the degrees of the numerator and denominator. ...
- Determining the End Behavior of a Rational Function - Vocabulary and Equations. ...
- Example Problem 1: Determining the End Behavior of a Rational Function. ...
- Example Problem 2: Determining the End Behavior of a Rational Function. ...
How do you describe end behavior?
The end behavior of a function f describes the behavior of the graph of the function at the "ends" of the x-axis. In other words, the end behavior of a function describes the trend of the graph if we look to the right end of the x-axis (as x approaches +∞ ) and to the left end of the x-axis (as x approaches −∞ ).
How do you find the end behavior of a function?
To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as x gets very large or very small, so its behavior will dominate the graph.
How do you find the end behavior of a graph?
The end behavior of a polynomial function is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity. The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.
How do you describe the end behavior of a polynomial?
End behavior: The end behavior of a polynomial function describes how the graph behaves as x approaches ±∞. ± ∞ . We can determine the end behavior by looking at the leading term (the term with the highest n -value for axn a x n , where n is a positive integer and a is any nonzero number) of the function.
How do you state the end behavior of a function?
The end behavior can be stated by specifying whether the function f(x) approaches plus or minus infinity, or a specific value, as x approaches plus...
How do you find the end behavior of a power function?
The end behavior of a power function is determined by whether the power is even or odd, and the sign of its coefficient. There are four possible c...
What does it mean to describe the end behavior?
Describing the end behavior of a function involves specifying what happens to the function's value as the input variable becomes large in size, eit...
What is the end behavior of a parabola?
A parabola will approach +infinity if its leading coefficient is positive, and -infinity is the leading coefficient is negative, as x approaches ei...
What is the end behavior of a graph?
Every graph has certain end behavior characteristics. The end behavior of a graph is defined as what is going on at the ends of each graph. In other words, in what direction are the ends of the graphs heading?
Why do the ends of a function go down?
The end behavior of the functions are all going down at both ends. This is because the leading coefficient is now negative . So, when you have a function where the leading term is negative with an even exponent, the ends of the function both go down. You can remember it this way:
What happens when a leading term is negative?
Let's take a look: Notice that when we have a leading term that is negative and has an odd exponent, the end behavior is reversed; the left side goes up, while the right side goes down. This is directly opposite from when we have a positive leading term with an odd exponent. You can remember it this way:
What happens when you have a negative exponent?
When you have a negative/even exponent leading term, the graph is 'sad' or frowning (down/down). Positive Leading Term with an Odd Exponent. Let's look at what happens when a function has a positive leading term with an odd exponent. Here are three function graphs with odd exponents on the leading term.
What is the End Behavior of a Function?
The end behavior of a function {eq}f (x) {/eq} refers to how the function behaves when the variable {eq}x {/eq} increases or decreases without bound. In other words, the end behavior describes the ultimate trend in the graph of {eq}f (x) {/eq} as we move towards the far right or far left of the {eq}x {/eq}-axis.
How to Find the End Behavior of a Function
Finding the end behavior of a function requires determining what happens to its formula when the variable takes very large (positive or negative) values. It is very helpful to understand the behavior of the different types of basic functions, such as polynomials.
Polynomials
Polynomial functions consist of sums or differences of powers. Any polynomial in the variable {eq}x {/eq} can be expressed in the form
