Understanding how the zeros of a polynomial function affect its graph is really important in Algebra II. Zeros, also called roots, help define how the graph looks, how it behaves at the ends, and where it changes direction.

Zeros of a Polynomial

1. What are Zeros?
   The zeros of a polynomial function, like ( f(x) ), are the ( x ) values that make ( f(x) = 0 ). If a polynomial has a degree of ( n ), it can have up to ( n ) real zeros.

2. How to Find Zeros?
   You can find zeros using different methods like factoring, synthetic division, or the Rational Root Theorem. For instance, in the polynomial ( f(x) = x^3 - 6x^2 + 11x - 6 ), the zeros are ( x = 1 ), ( x = 2 ), and ( x = 3 ).

Behavior at Zeros

1. Multiplicity of Zeros
   The way the graph behaves at a zero depends on its multiplicity:
   • Odd Multiplicity: If a zero has an odd multiplicity (like 1 or 3), the graph will cross the x-axis at that zero. For example, for ( g(x) = (x-2)(x^2-4) ), when ( x = 2 ) is a zero, the graph crosses the x-axis at the point ( (2, 0) ).
   • Even Multiplicity: If a zero has an even multiplicity (like 2 or 4), the graph will touch the x-axis but not cross it. For instance, in ( h(x) = (x-3)^2(x-5) ), when ( x = 3 ) is a zero of multiplicity 2, the graph touches the x-axis at ( (3, 0) ) and turns back.

Graph Shapes and Turning Points

1. Overall Shape:
   • Degree of Polynomial: The degree of a polynomial affects its shape. A polynomial with degree ( n ) can have a maximum of ( n-1 ) turning points. If the polynomial’s degree is even, the ends of the graph will both go up or both go down as ( x ) gets very large or very small. If the degree is odd, the ends will go in opposite directions.
2. Turning Points:
   • Turning points are where the graph changes direction. They usually appear between zeros and at the highest and lowest points.
   • For ( f(x) = x^4 - 4x^2 ), the zeros are ( x = -2, 0, 2 ). The turning points will be found in the spaces between these zeros and also outside them.

End Behavior

1. What Affects End Behavior?
   • The leading coefficient (the number in front of the highest degree term) and the degree of the polynomial determine how the graph behaves at the ends.
   • For an even degree with a positive leading coefficient, as ( x ) goes to positive or negative infinity, ( f(x) ) goes to infinity. On the other hand, with a negative leading coefficient, as ( x ) goes to positive or negative infinity, ( f(x) ) goes to negative infinity.
   • For an odd degree with a positive leading coefficient, as ( x ) goes to negative infinity, ( f(x) ) goes to negative infinity, while it goes to positive infinity as ( x ) goes to positive infinity. The opposite happens with a negative leading coefficient.

Summary

In short, knowing the zeros of a polynomial function is very important for accurately graphing these functions. The multiplicity of the zeros tells us whether the graph will cross or just touch the x-axis. The degree of the polynomial shows how many turning points there are and affects the overall end behavior of the graph, telling us if it goes up or down to infinity. Recognizing these patterns helps us predict the shape and behavior of polynomial graphs in algebra.