MAT 150 Module 5 – Polynomial Functions Lesson 1 – Properties and Graphs of Polynomial Functions d%20Nicole%20Polynomial.JPG

Standard Form of a Polynomial Function

Requirements to call a function a polynomial: All the powers of x must be positive (no negative powers or x terms in the denominator of a fraction) All the powers of x must be integers The coefficients of x are any real numbers The graph must be a smooth, continuous curve (no holes or jumps in the graph).

Example 1 Identify the degree and leading coefficient of the polynomials.

Example 1 - Solution Identify the degree and leading coefficient of the polynomials. This polynomial is already written in standard form because the exponents are in descending order. The degree is three and the leading coefficient is -3. A polynomial with degree three is called a cubic function.

Example 1 - Solution Identify the degree and leading coefficient of the polynomials. This polynomial is not written in standard form because the exponents are not in descending order. In standard form the function is The degree is five and the leading coefficient is 4.

Example 1 - Solution

The graph of a polynomial We can tell what the basic shape of the graph of a polynomial function will be without actually graphing.. To tell the basic shape of the graph of a polynomial function, we only need two pieces of information: Whether the degree is even or odd Whether the leading coefficient is positive or negative

End Behavior The end behavior of a polynomial refers to what happens to the y values when x approaches positive and negative infinity. In other words, it we zoom out and look at the graph, in which direction is the graph moving for large and small x values?

End Behavior The end behavior depends on two things: whether the degree of the polynomial is even or odd, and whether the leading coefficient is positive or negative. There are four possibilities.

End Behavior degreeanan end behaviorpictures evenpositiveUp on both sides evennegative Down on both sides oddpositive Up on the right, down on the left oddnegativeUp on the left, down on the right

Turning points The turning points of a polynomial are the points where the graph changes direction. The maximum number of turning points is the degree of the polynomial minus one.

Turning points Local Maxima Local Minimum This function has three turning points: two maxima and one minimum. The degree of the function must be at least four.

Example 2 For the following polynomials: I.Identify the degree and leading coefficient II.Decide whether the polynomial is cubic or quartic III.Identify the end behavior IV.Graph on the given window V.Identify any local maxima or minima VI.Identify the zeros (x-intercepts) from the graph if possible. a)f(x) = x 3 + 3x 2 – 4x. Window: [-10,10] by [-5, 15], xscl = 1, yscl = 1 b) f(x) = 2x 4 +5x 3 + 7x x +15. Window: [-5, 5] by [-50, 50], xscl = 1, yscl = 10

Example 2 - Solution a)f(x) = x 3 + 3x 2 – 4x. Window: [-10,10] by [-15, 15], xscl = 1, yscl = 1 Degree: 3 Leading Coefficient: 1 The function is cubic. End Behavior:

Example 2 - Solution

b) f(x) = 2x 4 +5x 3 + 7x x +15. Window: [-5, 5] by [-50, 50], xscl = 1, yscl = 10 Degree: 4 Leading Coefficient: 2 The function is quartic End Behavior:

Example 2 - Solution

Example 3 - Application A business has revenue given by the function R(x) = -3x 4 +28x x 2 -60x. a.Graph the function on the window: [0,15] by [0, 10000], xscl = 1, yscl = 1,000 b.What is the maximum revenue? How many units must be sold to achieve the maximum revenue?

Example 3 - Solution