Questions: Determine if the graph can represent a polynomial function. If so, assume that the end behavior and all turning points are represented in the graph. (a) Determine the minimum degree of the polynomial. (b) Determine whether the leading coefficient is positive or negative based on the end behavior and whether the degree of the polynomial is odd or even. (c) Approximate the real zeros of the function, and determine if their multiplicities are even or odd.

Determine if the graph can represent a polynomial function. If so, assume that the end behavior and all turning points are represented in the graph.
(a) Determine the minimum degree of the polynomial.
(b) Determine whether the leading coefficient is positive or negative based on the end behavior and whether the degree of the polynomial is odd or even.
(c) Approximate the real zeros of the function, and determine if their multiplicities are even or odd.

Transcript text: Determine if the graph can represent a polynomial function. If so, assume that the end behavior and all turning points are represented in the graph. (a) Determine the minimum degree of the polynomial. (b) Determine whether the leading coefficient is positive or negative based on the end behavior and whether the degree of the polynomial is odd or even. (c) Approximate the real zeros of the function, and determine if their multiplicities are even or odd.

Solution

Solution Steps

Step 1: Determine the minimum degree of the polynomial

To determine the minimum degree of the polynomial, count the number of turning points in the graph. The number of turning points is one less than the degree of the polynomial. The graph has 4 turning points, so the minimum degree of the polynomial is 5.

Step 2: Determine the sign of the leading coefficient

Examine the end behavior of the graph. The graph falls to negative infinity on both ends, indicating that the leading coefficient is negative. Since the polynomial degree is odd (5), the leading coefficient must be negative.

Step 3: Approximate the real zeros and their multiplicities

Identify the x-intercepts (real zeros) of the graph. The graph crosses the x-axis at approximately -5, -2, 1, and 3. Since the graph crosses the x-axis at each zero, the multiplicities of these zeros are odd.