Polynomial Function Characteristics to Know for Honors Pre-Calculus

Understanding polynomial functions is key in Honors Pre-Calculus. These functions have unique characteristics, like degree, leading coefficient, and zeros, which shape their graphs. Grasping these concepts helps predict graph behavior and lays the groundwork for calculus.

1. Degree of a polynomial

   • The degree is the highest exponent of the variable in the polynomial.
   • It determines the overall shape and behavior of the graph.
   • Polynomials can be classified as linear (degree 1), quadratic (degree 2), cubic (degree 3), etc.

2. Leading coefficient

   • The leading coefficient is the coefficient of the term with the highest degree.
   • It influences the direction of the graph as x approaches positive or negative infinity.
   • A positive leading coefficient indicates the graph rises to the right, while a negative one indicates it falls to the right.

3. End behavior

   • End behavior describes how the graph behaves as x approaches positive or negative infinity.
   • It is determined by the degree and leading coefficient of the polynomial.
   • For even degrees, both ends of the graph will either rise or fall together; for odd degrees, they will rise on one end and fall on the other.

4. Zeros/roots

   • Zeros (or roots) are the x-values where the polynomial equals zero.
   • They represent the points where the graph intersects the x-axis.
   • The number of zeros can be equal to the degree of the polynomial, but some may be complex or repeated.

5. Multiplicity of zeros

   • The multiplicity refers to how many times a particular zero occurs.
   • A zero with an odd multiplicity will cross the x-axis, while one with an even multiplicity will touch the x-axis and turn around.
   • Understanding multiplicity helps predict the behavior of the graph near the zeros.

6. Turning points

   • Turning points are points where the graph changes direction from increasing to decreasing or vice versa.
   • The maximum number of turning points is one less than the degree of the polynomial.
   • They are important for understanding the overall shape of the graph.

7. Local maxima and minima

   • Local maxima are the highest points in a particular interval, while local minima are the lowest points.
   • These points occur at turning points and are critical for analyzing the graph's behavior.
   • They help identify the range of the polynomial function.

8. y-intercept

   • The y-intercept is the point where the graph intersects the y-axis (when x = 0).
   • It can be found by evaluating the polynomial at x = 0.
   • The y-intercept provides a starting point for graphing the polynomial.

9. Symmetry

   • A polynomial function can exhibit symmetry about the y-axis (even functions) or the origin (odd functions).
   • Even functions have only even-degree terms and are symmetric about the y-axis.
   • Odd functions have only odd-degree terms and exhibit rotational symmetry about the origin.

10. Continuity and differentiability

    • Polynomial functions are continuous everywhere, meaning there are no breaks or holes in the graph.
    • They are also differentiable everywhere, allowing for the calculation of slopes and rates of change.
    • This property is essential for applying calculus concepts to polynomial functions.