Learning objective
End behavior is a concept in mathematics that helps us understand what a polynomial function does when the input values increase or decrease without bound. In other words, it describes how the function behaves as we move towards positive or negative infinity on the x-axis. When input values increase or decrease without bound, the leading term (the term with the highest degree) dominates the values of all lower-degree terms in the polynomial function. This dominance of the leading term is crucial because it ultimately determines the end behavior of the function.
The degree and sign of the leading term play a significant role in defining the end behavior of a polynomial function. To represent the end behavior as input values increase without bound, we use the notation lim_{x\to \infty} p(x) = \infty or lim_{x\to \infty} p(x) = -\infty. This notation helps us understand whether the function approaches positive infinity or negative infinity as the input values approach positive infinity.
Similarly, when we want to describe the end behavior as input values decrease without bound, we use the notation lim_{x\to -\infty} p(x) = \infty or lim_{x\to -\infty} p(x) = -\infty. This notation indicates whether the function approaches positive infinity or negative infinity as the input values approach negative infinity.
By analyzing the degree and sign of the leading term and understanding these notations, we can effectively describe and interpret the end behavior of polynomial functions.
Degree | Leading Coefficient | End Behavior | Graph of the function |
---|---|---|---|
\text{even} | \text{positive} | \lim_{x \to -\infty} f(x) \to + \infty \\ \lim_{ x \to +\infty} f(x) \to + \infty | \text{rises to the left and} \\ \text{to the right} |
\text{even} | \text{negative} | \lim_{x \to -\infty} f(x) \to - \infty \\ \lim_{ x \to +\infty} f(x) \to - \infty | \text{falls to the left and} \\ \text{to the right} |
\text{odd} | \text{positive} | \lim_{x \to -\infty} f(x) \to - \infty \\ \lim_{ x \to +\infty} f(x) \to + \infty | \text{falls to the left and} \\ \text{rises to the right} |
\text{odd} | \text{negative} | \lim_{x \to -\infty} f(x) \to + \infty \\ \lim_{ x \to +\infty} f(x) \to - \infty | \text{rises to the left and} \\ \text{falls to the right} |
Describe the end behavior for each of the following functions.
p\left(x\right) = -2x^4 + 3x^3 - 5x^2 + x + 1.
f\left(t\right) = t^3 - 2t^2 + t - 1
h\left(k\right)=2k+4-\dfrac{1}{2}k^5
w\left(x\right)=x^3+2x^6
Consider the end behavior for a function p\left(x\right):
\lim_{x \to \infty} p(x) \to \infty \\ \lim_{ x \to -\infty} p(x) \to - \infty
Describe the end behavior of p\left(x\right) in words.
State whether p\left(x\right) has an even or odd degree. Explain.
State the sign of the leading coefficient of p\left(x\right). Explain.
Degree | Leading Coefficient | End Behavior | Graph of the function |
---|---|---|---|
\text{even} | \text{positive} | \lim_{x \to -\infty} f(x) \to + \infty \\ \lim_{ x \to +\infty} f(x) \to + \infty | \text{rises to the left and} \\ \text{to the right} |
\text{even} | \text{negative} | \lim_{x \to -\infty} f(x) \to - \infty \\ \lim_{ x \to +\infty} f(x) \to - \infty | \text{falls to the left and} \\ \text{to the right} |
\text{odd} | \text{positive} | \lim_{x \to -\infty} f(x) \to - \infty \\ \lim_{ x \to +\infty} f(x) \to + \infty | \text{falls to the left and} \\ \text{rises to the right} |
\text{odd} | \text{negative} | \lim_{x \to -\infty} f(x) \to + \infty \\ \lim_{ x \to +\infty} f(x) \to - \infty | \text{rises to the left and} \\ \text{falls to the right} |