Match the polynomial function with one of the graphs I-VI. Give reasons for your choice. S(x)=(1

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

Key Concepts

Degree of the Polynomial

The degree indicates the highest power of x in the function and informs the maximum number of turning points and the overall shape of the graph. In a degree six polynomial, we expect up to five turning points and a specific end behavior based on whether the degree is even or odd.

Leading Coefficient Test

The leading coefficient, combined with the degree, determines the end behavior of the polynomial. An even degree with a positive leading coefficient means that as x approaches both positive and negative infinity, the function will rise, which is a critical consideration when matching graphs.

Zeros and Multiplicity

Identifying the zeros (roots) of the polynomial and their multiplicities helps in understanding the x-intercepts and the nature of the graph at these points. A zero with even multiplicity, for example, results in the graph touching the x-axis without crossing, influencing the shape of the curve.

Symmetry

Polynomials that consist solely of even powers of x exhibit symmetry about the y-axis, known as even symmetry. Recognizing this symmetry aids in predicting the overall structure and appearance of the graph.

Turning Points and Local Extrema

The number and nature of turning points (points where the graph changes direction) are governed by the degree of the polynomial. This concept is essential in analyzing how the function's graph behaves between zeros and inflection points.

Transcript

00:04 All right.

00:06 We're going to discuss the curve that represents the function f of x or excuse me, s of x equals 1⁄2x to the 6 power minus 2x to the fourth.

00:23 And there are several things that we're going to talk about.

00:26 We're going to talk about roots and end behavior.

00:30 So to talk about end behavior, we need to talk about the degree of the polynomial.

00:42 In this case, the degree is.

00:44 Six, the highest exponent is six.

00:47 So it's a six degree polynomial.

00:50 It's an even degree, and it's got a positive lead coefficient.

00:57 So that tells me that as x is approaching positive infinity, y is also approaching positive infinity, as x is approaching negative infinity, y is approaching negative infinity.

01:11 In other words, as x goes to the left, y is increasing, it's doing whatever it's doing in the middle, and as y, x goes to the right, y is also increasing.

01:21 So both ends are going up.

01:23 The end of our graph is increasing in both directions.

01:27 So next we want to talk about x and y intercepts.

01:34 So if we want to find our roots or our zeros, we're going to want to let the function, 1 half, x to the 6 power minus 2x to the 4th equal 0.

01:48 And if we factor out a common factor here, we have a common factor of one -half x to the fourth power.

01:59 And if i divide out in a one -half -x -to -the -fourth power, i'm going to be left with x -squared, and then negative 2 divided by one -half is going to be minus 4, and then x -to -the -fourth divided by x -to -the -fourth is 1.

02:11 So we actually have one -half -x -to -the -fourth times the quantity x -squared minus 4.

02:17 Inside the parentheses there, that's the difference of two, squares.

02:21 So this will factor to x plus 2 times x minus 2.

02:28 So now we have our function factored completely.

02:33 If we let each factor equal 0, we will have 1 half x to the 4th equal 0, x plus 2 equal to 0, and x minus 2 equal to 0.

02:51 If i multiply both sides by 2 on this first 0, i have x to the 4th, equals 0...

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