Polynomial Functions and Graphs

Higher Degree Polynomial Functions and Graphs an is called the leading coefficient n is the degree of the polynomial a0 is called the constant term Polynomial Function A polynomial function of degree n in the variable x is a function defined by where each ai is real, an  0, and n is a whole number.

Polynomial Function in General Form Polynomial Functions Polynomial Function in General Form Degree Name of Function 1 Linear 2 Quadratic 3 Cubic 4 Quartic Teachers: This definition for ‘degree’ has been simplified intentionally to help students understand the concept quickly and easily. The largest exponent within the polynomial determines the degree of the polynomial.

Maximum Number of Zeros: 0 Polynomial Functions f(x) = 3 ConstantFunction Degree = 0 Maximum Number of Zeros: 0

Maximum Number of Zeros: 1 Polynomial Functions f(x) = x + 2 LinearFunction Degree = 1 Maximum Number of Zeros: 1

Maximum Number of Zeros: 2 Polynomial Functions f(x) = x2 + 3x + 2 QuadraticFunction Degree = 2 Maximum Number of Zeros: 2

Maximum Number of Zeros: 3 Polynomial Functions f(x) = x3 + 4x2 + 2 Cubic Function Degree = 3 Maximum Number of Zeros: 3

Maximum Number of Zeros: 4 Polynomial Functions Quartic Function Degree = 4 Maximum Number of Zeros: 4

Leading Coefficient The leading coefficient is the coefficient of the first term in a polynomial when the terms are written in descending order by degrees. For example, the quartic function f(x) = -2x4 + x3 – 5x2 – 10 has a leading coefficient of -2.

The Leading Coefficient Test As x increases or decreases without bound, the graph of the polynomial function f (x) = anxn + an-1xn-1 + an-2xn-2 +…+ a1x + a0 (an ¹ 0) eventually rises or falls. In particular, For n odd: an > 0 an < 0 If the leading coefficient is positive, the graph falls to the left and rises to the right. If the leading coefficient is negative, the graph rises to the left and falls to the right. Rises right Falls left Falls right Rises left

The Leading Coefficient Test As x increases or decreases without bound, the graph of the polynomial function f (x) = anxn + an-1xn-1 + an-2xn-2 +…+ a1x + a0 (an ¹ 0) eventually rises or falls. In particular, For n even: an > 0 an < 0 If the leading coefficient is positive, the graph rises to the left and to the right. If the leading coefficient is negative, the graph falls to the left and to the right. Rises right Rises left Falls left Falls right

Example Use the Leading Coefficient Test to determine the end behavior of the graph of f (x) = x3 + 3x2 - x - 3. Falls left y Rises right x

Determining End Behavior Match each function with its graph. B. A. C. D.

Summary Look at the two graphs and discuss the questions given below. Graph B Graph A 1. How can you check to see if both graphs are functions? 2. How many x-intercepts do graphs A & B have? 3. What is the end behavior for each graph? 4. Which graph do you think has a positive leading coeffient? Why? 5. Which graph do you think has a negative leading coefficient? Why?