How does a constant or a coefficient affect the graph of a polynomial function? Please give a detailed response, this is for my notes on Polynomial Transformations, thank you for your help.

If the degree is odd and the lead coefficient is positive, then the right end of the graph will point up and the left end of the graph will point down. If the degree is odd and the lead coefficient is negative, then the right end of the graph will point down and the left end of the graph will point up. Polynomial Functions

Suppose your math teacher writes two functions and their graphs on the board.

The functions shown on the blackboard are examples of polynomial functions.
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A polynomial function is a function that is a sum of terms of the form axn, where a is a real number, x is a variable, and n is an integer, such that n ≥ 0.

As you may have noticed, though the two functions on the board are both polynomial functions, their graphs behave a bit differently. Notice that the ends of f(x) both point up, but in g(x), the left end points down and the right end points up (this is called the end behavior of a function). This difference in behavior is due to the degree and lead coefficient of each polynomial. The degree of a polynomial is the highest exponent, while the lead coefficient of the polynomial is the coefficient of the variable with the highest exponent.

In the equations shown on the blackboard, the highest exponent in f(x) is 4, so the degree of f(x) is 4, and the coefficient of x4 is 1, so the lead coefficient of f(x) is 1. Similarly, the degree of g(x) is 3, and the lead coefficient of g(x) is 1.

Now that we've gotten the vocabulary out of the way, let's take a closer look at how the values of the degree and lead coefficient affect the end behavior of a polynomial function.

End Behavior of a Polynomial Function
The end behavior of a polynomial is determined by its degree and lead coefficient and can be found using the following rules:

If the degree is even and the lead coefficient is positive, then both ends of the polynomial's graph will point up.
If the degree is even and the lead coefficient is negative, then both ends of the polynomial's graph will point down.
If the degree is odd and the lead coefficient is positive, then the right end of the graph will point up and the left end of the graph will point down.
If the degree is odd and the lead coefficient is negative, then the right end of the graph will point down and the left end of the graph will point up.
Notice that this explains the end behavior of the two polynomial functions on the chalkboard. As we said earlier, the degree of f(x) is 4, so it's even, and the lead coefficient is 1, so it's positive. According to our rules, both ends of f(x) should point up, which is exactly what happens in the graph.

As for g(x), the degree is 3, so it's odd, and the lead coefficient is 1, so it's positive. The rules tell us that the right end of the graph should point up, and the left end should point down. Once again, this is exactly what happens in the graph.