Leading Term

Definition

The leading term of a polynomial or power function is the term with the highest exponent. It is the term that dominates the behavior of the function as the input values become larger or smaller.

5 Must Know Facts For Your Next Test

1. The leading term of a polynomial or power function determines the long-term behavior of the function as the input values become very large or very small.
2. For a polynomial function, the leading term is the term with the highest degree, and it determines the end behavior of the function.
3. For a power function, the leading term is the term with the highest exponent, and it determines the rate of growth or decay of the function.
4. The coefficient of the leading term is also important, as it determines whether the function is increasing or decreasing as the input values become larger.
5. Identifying the leading term is crucial for sketching the graph of a polynomial or power function and understanding its key features, such as end behavior and rate of change.

Review Questions

• Explain how the leading term of a polynomial function affects the end behavior of the function.
  • The leading term of a polynomial function is the term with the highest degree, and it determines the end behavior of the function. If the leading term has a positive coefficient, the function will grow without bound as the input values become very large, and it will approach positive infinity. If the leading term has a negative coefficient, the function will decrease without bound as the input values become very large, and it will approach negative infinity. The degree of the leading term also determines the rate at which the function grows or decreases as the input values become very large.
• Describe how the leading term of a power function affects the rate of growth or decay of the function.
  • The leading term of a power function is the term with the highest exponent, and it determines the rate of growth or decay of the function. If the exponent of the leading term is positive, the function will grow at a rate determined by the value of the exponent. The larger the exponent, the faster the function will grow. If the exponent of the leading term is negative, the function will decay at a rate determined by the value of the exponent. The larger the absolute value of the negative exponent, the faster the function will decay.
• Analyze how the coefficient of the leading term of a polynomial or power function affects the function's behavior.
  • The coefficient of the leading term of a polynomial or power function is crucial in determining the function's behavior. If the coefficient is positive, the function will be increasing as the input values become larger. If the coefficient is negative, the function will be decreasing as the input values become larger. The magnitude of the coefficient also affects the rate of change of the function, with larger coefficients leading to faster rates of growth or decay. Understanding the role of the leading term coefficient is essential for sketching the graph of a polynomial or power function and predicting its key features, such as its end behavior and rate of change.

Related terms

Polynomial Function:

A polynomial function is a function that can be written in the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, where $a_n, a_{n-1}, ..., a_1, a_0$ are real numbers and $n$ is a non-negative integer.

Power Function:

A power function is a function that can be written in the form $f(x) = ax^b$, where $a$ and $b$ are real numbers and $b$ is not equal to 0.

Degree of a Polynomial: The degree of a polynomial is the highest exponent of the variable in the polynomial.