Graphing Polynomials

Definition

Graphing polynomials is the process of visually representing the behavior and characteristics of polynomial functions on a coordinate plane. This involves determining the key features of the polynomial, such as its degree, zeros, and end behavior, and then using that information to sketch an accurate graph of the function.

5 Must Know Facts For Your Next Test

1. The degree of a polynomial function determines the maximum number of turning points (local maxima and minima) in the graph.
2. The number of real zeros of a polynomial function is less than or equal to the degree of the polynomial.
3. The graph of a polynomial function is continuous and can be sketched by identifying the key features, such as the y-intercept, zeros, and end behavior.
4. Polynomial functions with even degrees have graphs that are either always increasing or always decreasing, while odd-degree polynomials have graphs that change direction.
5. The leading coefficient of a polynomial function determines the direction of the graph's end behavior, with a positive leading coefficient resulting in a graph that opens upward and a negative leading coefficient resulting in a graph that opens downward.

Review Questions

• Explain how the degree of a polynomial function affects the number of turning points in its graph.
  • The degree of a polynomial function determines the maximum number of turning points (local maxima and minima) in the graph. Specifically, a polynomial function of degree $n$ can have at most $n-1$ turning points. This is because the graph of a polynomial function is a continuous curve, and the number of times it can change direction is one less than the degree of the polynomial.
• Describe the relationship between the number of real zeros of a polynomial function and its degree.
  • The number of real zeros of a polynomial function is less than or equal to the degree of the polynomial. This is because the Fundamental Theorem of Algebra states that a polynomial function of degree $n$ has $n$ complex roots, which may include real and imaginary roots. However, if a polynomial function has complex roots, it will also have a corresponding real root, so the number of real roots will be less than or equal to the degree of the polynomial.
• Analyze how the leading coefficient of a polynomial function affects the end behavior of its graph.
  • The leading coefficient of a polynomial function determines the direction of the graph's end behavior. If the leading coefficient is positive, the graph of the polynomial function will open upward as the independent variable approaches positive or negative infinity. Conversely, if the leading coefficient is negative, the graph will open downward as the independent variable approaches positive or negative infinity. This is because the term with the highest degree will dominate the behavior of the function as the independent variable becomes very large or very small in magnitude.

Related terms

Polynomial Function: A polynomial function is a function that can be expressed as the sum of a finite number of non-negative integer powers of the independent variable, with coefficients.

Zeros of a Polynomial: The zeros of a polynomial function are the values of the independent variable for which the function equals zero, or the x-intercepts of the graph.

End Behavior: The end behavior of a polynomial function refers to how the graph of the function behaves as the independent variable approaches positive or negative infinity.