Chapter 2.2: End Behavior of Polynomial Functions

End Behavior: The Behavior of a graph as the graph of f(x) as x approaches infinity or negative infinity.

Polynomial functions with even degrees (ex.  x²+8x-15) approach either infinity or negative infinity on both ends of the graph, but never both.

Polynomials with odd degrees (ex. 2x³+5x²-13x+12) approach infinity on one end of the graph and negative infinity on the other. 

• Whether the function with the largest degree is positive or negative determines which way the ends of the graph will behave. 
• If a function with an even degree is negative, the ends will both approach negative infinity.

Example: Graph of f(x)=-x²-6x+8

• If a function with an odd degree is negative, the graph will approach infinity on the left side and approach negative infinity on the right.

The reason that the first degree determines the behavior of the graph is because, during End Behavior, when x approaches infinity or negative infinity, it overshadows all lesser values due to having the largest exponent.

As \[x\rightarrow \infty\] or -∞, f(x) approaches infinity as well.

Multiplicity

The zeroes of a graph of a polynomial function correspond to x-intercepts. Each intercept has a "multiplicity", or how many times it's associated factor appears in the polynomial. 

Ex. f(x)=(x+3)(x-2) has multiplicities of one -3 and one 2. f(x)=(x+3)(x-2)² has multiplicities of one -3 and two 2.

• The graph of function (x+6)(x-7) has 7 multiplicity 1 and -6 multiplicity 1.

• The graph of function (x+6)(x-7)(x-7) has -6 multiplicity 1 and 7 multiplicity 2.

Both functions have the same x-intercepts but have different multiplicities. The number of occurrences of the intercepts determine whether the line crosses the x-axis or turns away from it.

If the multiplicity is greater than 1, it will turn back.