Question

Is there a relationship between the degree of a polynomial and how "steep" it is on the left and right edges? If so, what is it

Answer 1

The degree of a polynomial determines the steepness at the edges of its graph; higher degrees lead to steeper slopes for large absolute values of the variable.

Step-by-step explanation:

Yes, there is a relationship between the degree of a polynomial and how steep it is on the left and right edges of its graph. The degree of the polynomial indicates the highest power of the variable in the polynomial equation. As the degree increases, the rate at which the function values (y-values) increase or decrease also grows, which results in steeper slopes on the edges of the graph.

For example, a first-degree polynomial, which is a linear function, has a graph that is a straight line. It has a constant slope throughout its length. In contrast, a second-degree polynomial or quadratic function will have a parabola shape, which becomes steeper away from the vertex. As the degree of the polynomial increases, such as in cubic, quartic, or higher-degree polynomials, the ends of the graph become even steeper. This effect is particularly noticeable for large absolute values of the variable (x-values). In summary, the higher the degree of the polynomial, the more pronounced is the steepness at the extremities of its graph.

Answer 2

The larger the degree, the steeper the graph's branches towards the right and left edges.

Step-by-step explanation:

Yes, there is a relationship between the degree of a polynomial and how steep its branches are at their end behavior (for large positive values of x, and to the other end: towards very negative values of x).

This is called the "end behavior" of the polynomial function, and is dominated by the leading term of the polynomial, since at very large positive or very negative values of the variable "x" it is the term with the largest degree in the polynomial (the leading term) the one that dominates in magnitude over the others.

Therefore, larger degrees (value of the exponent of x) correspond to steeper branches associated with the geometrical behavior of "power functions" (functions of the form:

`f(x)=a_n\,x^n`

which have characteristic end behavior according to even or odd values of the positive integer "n").

Recalling the behavior of such power functions, the larger the power (the degree), the steeper the graph.