5-10= Match the polynomial function with one of the graphs I-VI. Give reasons for your choice. (

5-10= Match the polynomial function with one of the graphs...

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of the variable. Understanding the algebraic structure allows one to deduce properties about its graph, such as the number of turning points, behavior near its roots, and overall shape.

Degree and Leading Coefficient

The degree of a polynomial is the highest power of the variable present in the expression, which largely determines the end behavior of its graph. The leading coefficient (the coefficient of the term with the highest degree) further influences the direction in which the graph opens at the ends. For example, even-degree polynomials with a negative leading coefficient will tend to have both ends falling, while odd-degree polynomials have opposite end behaviors.

Zeros and Multiplicity

Zeros of the polynomial are the solutions to when the polynomial equals zero, and the multiplicity of a zero refers to the number of times a particular root appears. The multiplicity affects how the graph interacts with the x-axis—if the multiplicity is even, the graph will touch the axis and turn around at that zero, whereas if it’s odd, the graph will cross the axis.

Symmetry of Even Functions

An even function is symmetric about the y-axis, meaning that f(x) = f(-x) for all x in its domain. Recognizing even symmetry in a polynomial function can significantly simplify graph analysis because the behavior on one side of the y-axis mirrors the behavior on the other side.

Transcript

00:03 All right.

00:05 So we have the function, q of x equals negative x squared times x squared minus 4.

00:21 And we want to discuss the graph of this function.

00:24 So there's several things that we need.

00:26 We first of all need our intercepts, both x and y.

00:32 When we talk about the roots, we'll need to know the multiplicity.

00:35 So we can discuss the behavior of the graph at those roots.

00:39 We also need to discuss n behavior to help us determine what the graph's going to look like.

00:51 So right now, if i actually distribute that negative x squared, then we can see that q of x is equal to negative x to the fourth plus 4x squared.

01:04 This is important because our lead term right here is the negative x to the fourth.

01:10 It means it's a polynomial whose degree is 4, and it has a negative lead coefficient.

01:20 So that tells me that as x is approaching infinity, y will be approaching negative infinity.

01:29 And as x is approaching negative infinity, y is approaching negative infinity.

01:38 In other words, when we look at the graph of this thing, it's going to be going down to the left, whatever it's doing in the middle, and it's going to be going down to the right.

01:49 Okay, so that's the end behavior.

01:51 Now, to discuss the roots, if we let q of x equal zero, then that means negative x squared times x squared minus four is equal to zero.

02:11 So we have negative x squared minus four is the difference of two squares.

02:16 It will factor to x plus two times x minus 2.

02:21 So now we have all of our factors.

02:24 We let each factor equal 0...

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