Polynomial Functions and Graphs

Graphs of Polynomial Functions  

Graphs of polynomials are smooth and continuous. No gaps or holes, can be drawn without lifting pencil from paper No sharp corners or cusps This IS the graph of a polynomial This IS NOT the graph of a polynomial

A polynomial function is a function of the form: Remember integers are … –2, -1, 0, 1, 2 … (no decimals or fractions) so positive integers would be 0, 1, 2 … A polynomial function is a function of the form: n must be a positive integer All of these coefficients are real numbers The degree of the polynomial is the largest power on any x term in the polynomial.

A polynomial function is a function of the form: Notice the n values on x should be arranged in descending order. a0 is called the Constant term an is called the Leading Coefficient

Determine which of the following are polynomial functions Determine which of the following are polynomial functions. If the function is a polynomial, state its degree. A polynomial of degree 4. A polynomial of degree 0. x 0 We can write in an x0 since this = 1. Not a polynomial because of the square root since the power is NOT an integer Not a polynomial because of the x in the denominator since the power is negative

Polynomial Functions with special names 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.

Finding intercepts/solutions/zeros/factors of polynomials: A real number a is a zero of a function y = f (x) if and only if f (a) = 0. Real Zeros of Polynomial Functions If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent. 1. x = a is a zero of f. 2. x = a is a solution of the polynomial equation f (x) = 0. 3. (x – a) is a factor of the polynomial f (x). 4. (a, 0) is an x-intercept of the graph of y = f (x). A polynomial function of degree n has at most n zeros. Zeros of a Function

Polynomial Functions Example: f(x) = 3 *Constant Function *Degree: 0 *Maximum zeros: 0 *Leading Coeff: 3

* Maximum Number of Zeros: (Cont.) f(x) = x + 2 * Linear Function * Degree: 1 * Maximum Number of Zeros: 1 * Leading Coeff: 1

* Maximum Number of Zeros: (Cont.) f(x) = x2 + 3x + 2 * Quadratic Function * Degree: 2 * Maximum Number of Zeros: 2 * Leading Coeff: 1

* Maximum Number of Zeros: (Cont.) f(x) = x3 + 4x2 + 2 * Cubic Function * Degree: 3 * Maximum Number of Zeros: 3 * Leading Coeff: 1

The Leading Coefficient Test (for determining end behavior) 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 (odd degree): an > 0 an < 0 Increasing over entire domain Decreasing over entire domain 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 (even degree): 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.

Quartic Polynomials 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?

How to find x-Intercepts (real zeros).. Number Of x-Intercepts of a Polynomial Function A polynomial function of degree n will have a maximum of n x- intercepts (real zeros). Find all zeros of f (x) = -x4 + 4x3 - 4x2. -x4 + 4x3 - 4x2 = 0 We now have a polynomial equation. x4 - 4x3 + 4x2 = 0 Multiply both sides by -1. (optional step) x2(x2 - 4x + 4) = 0 Factor out x2. x2(x - 2)2 = 0 Factor completely. x2 = 0 or (x - 2)2 = 0 Set each factor equal to zero. x = 0 x = 2 Solve for x. x-intercepts: (0,0) (2,0)

Multiplicity and x-Intercepts If r is a zero of even multiplicity, then the graph touches the x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph crosses the x-axis at r. Regardless of whether a zero is even or odd, graphs tend to flatten out at zeros with multiplicity greater than one.

How to find x-Intercepts (real zeros).. Go back at example and look at multiplicity Find all zeros of f (x) = -x4 + 4x3 - 4x2. x2(x - 2)2 = 0 Factor completely. x2 = 0 or (x - 2)2 = 0 Set each factor equal to zero. Even Multiplicity of 2 for each of these zeros x = 0, x = 2 **So, at the x-intercepts: (0,0) and (2,0) the graph** will turn at these points.

Example: Find all the real zeros of f (x) = x 4 – x3 – 2x2. Factor completely: f (x) = x 4 – x3 – 2x2 = x2(x + 1)(x – 2). y x –2 2 f (x) = x4 – x3 – 2x2 The real zeros are x = –1(Mult of 1), x = 0 (Mult of 2) , x = 2. (Mult of 1), (–1, 0) (0, 0) (2, 0) These correspond to the x-intercepts. Example: Real Zeros

Example: Find the x-intercepts and multiplicity of f(x) =2(x+2)2(x-3)  

Additional Helpful information for graphing polynomials Turning points – where the graph of a function changes from increasing to decreasing or vice versa. The number of turning points of the graph of a polynomial function of degree n  1 is at most n – 1. EXAMPLE: Each Turning point results in local extrema (local max or minimum) f (x) = x4 – x3 – 2x2

Number of Local Extrema A linear function has degree 1 and no local extrema. A quadratic function has degree 2 with one extreme point. A cubic function has degree 3 with at most two local extrema. A quartic function has degree 4 with at most three local extrema.

Steps for sketching the graph of a polynomial by hand: Apply the leading Coeff Test Determine the maximum turning points Find the real zeros by setting the function equal to zero and solving through factoring. Consider multiplicity Determine a few additional points using a t-chart Draw the graph with a continuous line through the points

Example: