Match the polynomial function with one of the graphs I-VI. Give reasons for your choice. R(x)=-x

Match the polynomial function with one of the graphs I-VI....

Key Concepts

Polynomial Functions

Polynomial functions are expressions constructed from variables and coefficients using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Their graphs are continuous and smooth, which makes analyzing key features like intercepts, turning points, and end behavior straightforward.

Degree and Leading Coefficient

The degree of a polynomial (the highest power of x) and its leading coefficient largely determine the overall shape and behavior of the graph. For instance, polynomials with odd degrees have end behaviors where the two ends of the graph head in opposite directions, and a negative leading coefficient reverses the typical up/down orientation.

Zeros of the Polynomial

The zeros of a polynomial are the x-values for which the polynomial equals zero, indicating where the graph crosses or touches the x-axis. Analyzing and factoring the polynomial to determine its zeros is crucial for sketching an accurate graph, as these points form the backbone of the function's graph.

End Behavior Analysis

End behavior analysis involves examining how the value of the polynomial function changes as x approaches positive or negative infinity. This understanding is guided by the leading term and provides insight into how the graph will extend beyond the local features, thereby helping in matching the function to its corresponding graph.

Transcript

00:03 We want to discuss the graph of r of x equals negative x to the fifth plus 5x cubed minus 4x.

00:22 There are several things we have to discuss.

00:24 We have to discuss the roots, x and y intercepts.

00:27 We want to discuss the degree of the polynomial and end behavior and possibly multiplicity of the roots.

00:34 So that way we can get an idea of.

00:38 Of what is going on with this curve.

00:43 So the first thing i notice is this is a fifth degree polynomial.

00:49 So it's an odd degree and the lead coefficient is negative.

00:54 What that tells us is that as x goes to the left towards negative infinity, y will approach positive infinity.

01:08 In other words, as x is decreasing, y is increasing.

01:12 And as x approaches positive infinity, y is approaching negative infinity.

01:19 In other words, as x is increasing, y is decreasing.

01:23 So it means we're going up to the left.

01:26 It's doing whatever it's doing in the middle, and it's going down to the right.

01:32 So that tells us the end behavior.

01:34 So it gives us an idea of which way our graphs going.

01:37 Next, we're going to talk about our x intercepts or our zeros.

01:41 So if we let negative x to the fifth power plus 5x cubed minus 4x equals zero, we can solve this for x.

01:56 I'm going to start by factoring out a common factor of negative x, and that'll leave us with x to the fourth minus 5x squared plus 4 equals 0.

02:13 And right now i'll have negative x times now.

02:19 I can factor this like i would factor our quadratic.

02:22 I'm going to have x squared minus 4 times x squared minus 1 equals 0.

02:29 Right, negative 4 times 1 is positive 4...

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