End Behavior

Definition

The end behavior of a function refers to the behavior or pattern of the function as it approaches positive or negative infinity. It describes the long-term trends and asymptotic properties of the function, providing insights into how the function will continue to behave outside of the given domain.

5 Must Know Facts For Your Next Test

1. The end behavior of a polynomial function is determined by the degree and sign of the leading coefficient.
2. For a power function $f(x) = x^n$, the end behavior is positive infinity as $x \rightarrow +\infty$ and negative infinity as $x \rightarrow -\infty$ if $n$ is even, and negative infinity as $x \rightarrow +\infty$ and positive infinity as $x \rightarrow -\infty$ if $n$ is odd.
3. Rational functions have two types of end behavior: the end behavior of the numerator and the end behavior of the denominator, which can be determined using the degree and sign of the leading coefficients.
4. The end behavior of an exponential function $f(x) = a^x$ is positive infinity as $x \rightarrow +\infty$ and 0 as $x \rightarrow -\infty$ if $a > 1$, and 0 as $x \rightarrow +\infty$ and positive infinity as $x \rightarrow -\infty$ if $0 < a < 1$.
5. The end behavior of a function can be used to sketch its graph and understand its long-term trends, which is crucial for analyzing the behavior of polynomial, rational, and exponential functions.

Review Questions

• Explain how the degree and sign of the leading coefficient affect the end behavior of a polynomial function.
  • The end behavior of a polynomial function is determined by the degree and sign of the leading coefficient. If the leading coefficient is positive, the function will approach positive infinity as $x \rightarrow +\infty$ and negative infinity as $x \rightarrow -\infty$. If the leading coefficient is negative, the function will approach negative infinity as $x \rightarrow +\infty$ and positive infinity as $x \rightarrow -\infty$. The degree of the polynomial determines the rate at which the function approaches these asymptotic values.
• Describe the relationship between the end behavior of a rational function and the degrees of the numerator and denominator polynomials.
  • The end behavior of a rational function $R(x) = \frac{P(x)}{Q(x)}$ is determined by the degrees of the numerator polynomial $P(x)$ and the denominator polynomial $Q(x)$. If the degree of $P(x)$ is greater than the degree of $Q(x)$, then the function will approach positive or negative infinity as $x \rightarrow +\infty$ or $x \rightarrow -\infty$, depending on the signs of the leading coefficients. If the degree of $P(x)$ is less than the degree of $Q(x)$, then the function will approach 0 as $x \rightarrow +\infty$ or $x \rightarrow -\infty$. If the degrees are equal, the end behavior is determined by the ratio of the leading coefficients.
• Analyze how the end behavior of an exponential function $f(x) = a^x$ changes as the base $a$ varies.
  • The end behavior of an exponential function $f(x) = a^x$ is determined by the value of the base $a$. If $a > 1$, the function will approach positive infinity as $x \rightarrow +\infty$ and 0 as $x \rightarrow -\infty$. If $0 < a < 1$, the function will approach 0 as $x \rightarrow +\infty$ and positive infinity as $x \rightarrow -\infty$. This is because the exponential function grows or decays exponentially, with the rate of growth or decay determined by the base $a$. Understanding the end behavior of exponential functions is crucial for analyzing their long-term trends and asymptotic properties.

Related terms

Asymptote: A straight line that a function's graph approaches but never touches as the function's input values approach positive or negative infinity.

Polynomial Function: A function that can be expressed in the form $P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, where $a_n \neq 0$ and $n$ is a non-negative integer.

Rational Function: A function that can be expressed as the ratio of two polynomial functions, $R(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$.