Problem 83 - Algebra and Trigonometry

Question

Explain the relationship between the multiplicity of a zero and whether or not the graph crosses or touches the x-axis and turns around at that zero.

Step-by-Step Solution

Answer

Roots of odd multiplicity make the graph cross the x-axis, while roots of even multiplicity make the graph touch the x-axis and turn around at that point.

Step 1: Introduction to Multiplicity of Zeros

A zero of a function is called a root or a solution. The multiplicity of a root is essentially the number of times that the root appears, or equivalently the power or exponent of the root in the factored form of the function.

Step 2: Multiplicity and The Graph

The multiplicity of a zero has a specific correlation to how the function interacts with the x-axis. If the root has an odd multiplicity, then the graph will cross the x-axis through that zero. If the root has an even multiplicity, then the graph will touch the x-axis at that zero, but will not cross it, meaning it turns around at that point.

Step 3: Example

Consider a function \( f(x) = (x-1)^2(x+2) \), it has zeros at x=1 and x=-2. At x=1 the graph touches the x-axis and turns around because 1 is a root of even multiplicity (2). At x=-2 the graph crosses the x-axis because -2 is a root of odd multiplicity (1).

Key Concepts

Graphing Polynomial Functions

Understanding how to graph polynomial functions is critical for visualizing the behavior of these equations. To graph a polynomial function, we plot a curve by calculating a few essential points and displaying the overall shape.

For starters, identify the zeros of the function, which are the values of 'x' where the function equals zero. Plot these points on the x-axis. Then, determine the end behavior by observing the leading coefficient and the degree of the polynomial: if the degree is even, the ends will face in the same direction; if it's odd, they'll face opposite directions. Using these points, along with any given maxima or minima and the y-intercept, you can sketch the curve, connecting these characteristics with a smooth continuous line.

Zero of a Function

When we talk about the 'zero of a function,' we refer to the value or values of 'x' which make the function's value equal to zero. Simply put, these are the roots of the equation. Finding these zeros is a foundational skill for analyzing the function's behavior.

For example, in a quadratic equation like \( f(x) = x^2 - 4 \), the zeros are the solutions to \( x^2 - 4 = 0 \), which are \( x = 2 \) and \( x = -2 \). These serve as critical points where the graph of the function intersects the x-axis.

Root Multiplicity and Graph Behavior

Root multiplicity directly affects the curvature of the graph near the x-axis. In essence, it tells us whether the graph merely touches the axis or actually crosses it at the root's location.

Each root of a function can have a multiplicity, which is the number of times that particular root appears when the function is factored. For instance, in the polynomial \( f(x) = (x-3)^3 \), the root x=3 has a multiplicity of 3, indicating that the graph will behave in a certain way around this point. Specifically, a root with odd multiplicity will result in the graph crossing the x-axis, while a root with even multiplicity will mean the graph touches and turns back, making a local maximum or minimum at that point.

Even and Odd Multiplicity

The concept of even and odd multiplicity is pivotal in predicting how the graph of a polynomial behaves at its roots. An even multiplicity means that the graph touches the x-axis at the zero and turns around. This gives the characteristic 'U' shape at the root. An odd multiplicity, however, implies that the graph will pass through the x-axis at the zero.

As a visual example, the graph of a function like \( f(x) = (x-2)^2 \) at \( x = 2 \) would display an even multiplicity since the root is squared. The graph approaches this point, touches the x-axis, and then reverses direction. Meanwhile, a root with an exponent of 1, 3, 5, etc., exhibits odd multiplicity, which means the graph will cross the x-axis at these points.