Finding The Zeros Of Function Y=(x-4)(x^2-12x+36)
In the realm of mathematics, finding the zeros of a function is a fundamental task that unlocks critical insights into its behavior and properties. Zeros, also known as roots or x-intercepts, are the values of the independent variable (typically 'x') that make the function's output (typically 'y') equal to zero. In simpler terms, they are the points where the graph of the function intersects the x-axis. This article delves into the process of determining the zeros of a specific function, providing a step-by-step explanation and highlighting the underlying mathematical principles. We will dissect the function $y=(x-4)(x^2-12x+36)$, meticulously unraveling its structure to identify the values of 'x' that render the function's output zero. This exploration will not only equip you with the ability to solve similar problems but also deepen your understanding of the crucial role zeros play in analyzing and interpreting mathematical functions.
Understanding Zeros of a Function
Before diving into the specifics of our function, let's solidify our understanding of what zeros truly represent. The zeros of a function are the solutions to the equation f(x) = 0, where f(x) represents the function. Geometrically, these zeros correspond to the points where the graph of the function crosses or touches the x-axis. Identifying zeros is essential for several reasons:
- Graphing: Zeros provide key anchor points for sketching the graph of a function. Knowing where the graph intersects the x-axis allows us to accurately map its behavior.
- Solving Equations: Finding the zeros of a function is equivalent to solving the equation f(x) = 0. This is a fundamental skill in various mathematical contexts.
- Analyzing Behavior: The zeros of a function often reveal critical information about its behavior, such as intervals where the function is positive or negative, and potential turning points.
Deconstructing the Function: y=(x-4)(x^2-12x+36)
Now, let's turn our attention to the function at hand: $y=(x-4)(x^2-12x+36)$. This function is expressed in a factored form, which is a significant advantage when determining its zeros. The function is a product of two factors: (x-4) and (x^2-12x+36). To find the zeros, we need to identify the values of 'x' that make either of these factors equal to zero. This is based on the zero-product property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero.
Factor 1: (x-4)
The first factor, (x-4), is a simple linear expression. To find the value of 'x' that makes this factor zero, we set it equal to zero and solve:
Adding 4 to both sides of the equation, we get:
Thus, the first zero of the function is x = 4. This means that the graph of the function intersects the x-axis at the point (4, 0).
Factor 2: (x^2-12x+36)
The second factor, (x^2-12x+36), is a quadratic expression. To find its zeros, we need to solve the quadratic equation:
There are several methods to solve quadratic equations, including factoring, using the quadratic formula, or completing the square. In this case, the quadratic expression is a perfect square trinomial, which means it can be factored easily. A perfect square trinomial is a trinomial that can be expressed as the square of a binomial. Recognizing this pattern allows us to simplify the solution process.
Factoring the Quadratic Expression
The quadratic expression (x^2-12x+36) can be factored as follows:
Now, we have the factored form of the quadratic expression. To find the zeros, we set this factored form equal to zero:
Taking the square root of both sides, we get:
Adding 6 to both sides, we find:
Therefore, the second zero of the function is x = 6. Since the factor (x-6) appears twice in the factored form of the function, we say that x = 6 is a zero with multiplicity 2. This means that the graph of the function touches the x-axis at the point (6, 0) but does not cross it. The concept of multiplicity is crucial for understanding the behavior of a function's graph near its zeros.
Consolidating the Zeros
Having analyzed both factors of the function, we have identified the following zeros:
- x = 4 (from the factor (x-4))
- x = 6 (from the factor (x^2-12x+36), with multiplicity 2)
Therefore, the zeros of the function $y=(x-4)(x^2-12x+36)$ are 4 and 6. It's important to note that while 6 is a zero with multiplicity 2, it is still considered a zero of the function.
Selecting the Correct Answer
Now that we have determined the zeros of the function, let's revisit the original question and the answer choices:
What are the zeros of the function $y=(x-4)(x^2-12x+36) ?$
A. -6, 4, and 6 B. 4 and 6 C. -6 and -4 D. 0, 4, and 6
Based on our analysis, the correct answer is B. 4 and 6. The other options include incorrect values or omit the correct zeros.
Visualizing the Function's Graph
To further solidify our understanding, let's briefly discuss the graphical representation of the function. The zeros we found, x = 4 and x = 6, correspond to the points where the graph intersects or touches the x-axis. Since x = 6 is a zero with multiplicity 2, the graph will touch the x-axis at (6, 0) but not cross it. At x = 4, the graph will cross the x-axis. By understanding the zeros and their multiplicities, we can gain valuable insights into the shape and behavior of the function's graph.
Conclusion
In this comprehensive exploration, we successfully determined the zeros of the function $y=(x-4)(x^2-12x+36)$. By applying the zero-product property and factoring techniques, we identified the zeros as 4 and 6. We also discussed the concept of multiplicity, which provides additional information about the function's behavior near its zeros. This process underscores the importance of understanding zeros in analyzing and interpreting mathematical functions. The ability to find zeros is a fundamental skill that empowers us to solve equations, graph functions, and gain deeper insights into their properties. This detailed walkthrough not only provides the solution to the specific problem but also serves as a guide for tackling similar challenges in the future. By mastering the techniques presented here, you will be well-equipped to unravel the mysteries of various mathematical functions and their zeros.
This exploration highlights the significance of understanding the relationship between algebraic expressions and their graphical representations. The zeros of a function, derived algebraically, directly translate to the x-intercepts on its graph. This connection allows us to visualize and interpret mathematical concepts in a more intuitive way. Furthermore, the concept of multiplicity adds another layer of understanding, revealing how the graph behaves near its zeros. A zero with multiplicity 2, as seen in our example, indicates a point where the graph touches the x-axis but doesn't cross it, signifying a turning point or a local extremum. Grasping these nuances is crucial for accurately sketching and analyzing functions.
In conclusion, the journey of finding the zeros of the function $y=(x-4)(x^2-12x+36)$ has been more than just a mathematical exercise. It has been an exploration of fundamental concepts, problem-solving strategies, and the interconnectedness of algebra and geometry. The zeros, 4 and 6, are not merely numerical solutions; they are keys that unlock a deeper understanding of the function's behavior and its place within the broader landscape of mathematics. By mastering these techniques and concepts, you are empowered to approach mathematical challenges with confidence and insight. The ability to identify and interpret zeros is a valuable asset in various fields, from engineering and physics to economics and computer science. It is a testament to the power of mathematics as a tool for understanding and shaping the world around us. So, continue to explore, question, and delve deeper into the fascinating world of mathematics, and you will discover the beauty and power that lies within its intricate web of concepts and applications.
