How does even multiplicity impact a graph?
It means the graph is symmetric about the y-axis, since f(-x) = f(x)
Sorry; I was thinking of even functions.
You know that when the graph crosses the x-axis, there's a root there. The function changes from - to +. However, a root of even multiplicity means that f(x) is positive on both sides of the root, since x^2>=0. So, the graph is tangent to the x-axis, rather than crossinng it.
The even multiplicity of a root in a polynomial equation has a significant impact on the graph of the equation.
To understand this impact, let's first define what even multiplicity means. In a polynomial equation, the multiplicity of a root refers to the number of times that root appears as a factor of the equation. A root with an even multiplicity means that it appears twice or more as a factor.
Now, let's consider the effects of even multiplicity on the graph of the equation:
1. Touches or bounces: If a root has an even multiplicity, the graph of the equation will touch or bounce off the x-axis at that root. The graph essentially behaves like it is "kissing" the x-axis without crossing it. This behavior occurs because the repeated factors associated with the even multiplicity cause the graph to change direction without crossing the x-axis.
2. Smoothness: When a root has an even multiplicity, the graph near that root is smooth. Unlike roots with odd multiplicities, which create a sharp bend or a cusp in the graph, even multiplicities result in a smooth curve around the root.
3. Turning points: The number of times a graph intersects the x-axis at a root is determined by the multiplicity of that root. For example, a root with even multiplicity will have one turning point if it appears twice, two turning points if it appears four times, and so on.
In summary, even multiplicities in a polynomial equation affect the graph by causing it to touch or bounce off the x-axis at the corresponding roots, creating a smooth curve around those roots, and influencing the number of turning points.