Problem 11 - Precalculus : Building Concepts and Connections

Question

Determine the multiplicities of the real zeros of the function. Comment on the behavior of the graph at the \(x\) -intercepts. Does the graph cross or just touch the \(x\) -axis? You may check your results with a graphing utility. $$f(x)=x^{2}+2 x+1$$

Step-by-Step Solution

Answer

The real zero of the function is -1 and its multiplicity is 2. The graph touches the x-axis at the point (-1,0).

Step 1: Identify the function

The function given is \(f(x) = x^2 + 2x + 1\).

Step 2: Factor the function

The function can be factored to \(f(x) = (x + 1)^2\).

Step 3: Find the roots

Setting the function equal to zero and solving for x gives \(x + 1 = 0 \rightarrow x = -1\).

Step 4: Identify the Multiplicity

Since the factor \(x + 1\) appears twice in the factored form of the equation, the multiplicty of the root \(x = -1\) is 2.

Step 5: Graph Behavior Analysis

Since the multiplicity of the root is even, the graph of the function touches the x-axis at the point \((-1,0)\) and then turns back the way it came without crossing.

Key Concepts

Factoring Quadratic Equations

Factoring quadratic equations is a common method used to solve quadratic functions. A quadratic equation generally takes the form:

• x^2 + bx + c = 0.

To factor a quadratic equation, we aim to express it as a product of simpler binomial expressions. In our case, given is the quadratic function:

• f(x) = x^2 + 2x + 1.

The goal is to rewrite this function in the form

• (x + p)(x + q).

For this particular equation, you can factor it as

• (x + 1)(x + 1), or (x + 1)^2.

This technique helps identify the solutions or roots of the equation by setting each factor to zero. Factoring is especially useful because it transforms a complicated equation into a simpler, more solvable form. Through factoring, you can directly see the values of x that satisfy the equation, known as the roots.

X-Intercepts Behavior

The behavior of x-intercepts helps us understand how a graph interacts with the x-axis. The x-intercepts of a function are the points where the graph crosses or touches the x-axis and are found by solving the equation

• f(x) = 0.

For the function

• f(x) = (x + 1)^2,

we see that it has a root at x = -1. This means that the graph touches the x-axis at the point (-1, 0). It's interesting to note that at this x-intercept, the graph touches but does not cross the x-axis. This occurrence can be attributed to the fact that the multiplicity of our x-intercept is even.
In summary, understanding x-intercepts provides valuable insights into the solutions of the function as well as the graph's behavior. Will the graph just touch, or will it cross the x-axis? The multiplicity of the roots helps determine this behavior, as detailed in the next section.

Even Multiplicity Effects

The concept of multiplicity is crucial in understanding the way graphs behave at their x-intercepts. When we talk about multiplicity, we refer to the number of times a particular root appears in the factorized form of a polynomial equation.
For the quadratic function

• f(x) = (x + 1)^2,

the root x = -1 appears twice, thus giving it a multiplicity of 2, which is an even number.
Whenever the multiplicity of a root is even, it affects the graphic representation at that x-intercept. Specifically, it causes the graph to merely touch the x-axis at the intercept point without crossing it. Essentially, the graph will come down towards the x-axis, make contact, and then retreat back in the direction it came from.

• This even multiplicity leads to the graph having a "bounce" effect at the intercept.
• It will not "cut" across the x-axis.

Understanding even multiplicity effects thus allows us to anticipate the graph’s behavior at its roots and provides a deeper comprehension of polynomial functions.