Axis Of Symmetry: F(x) = -(x + 9)(x - 21) Explained

Understanding the axis of symmetry is crucial for grasping the behavior and properties of quadratic functions. In this article, we'll delve into the function f(x) = -(x + 9)(x - 21) and explore how to determine its axis of symmetry. Whether you're a student tackling algebra or simply curious about mathematical concepts, this guide will provide a clear and comprehensive explanation. Let's unlock the secrets of this function together.

What is the Axis of Symmetry?

To fully understand the axis of symmetry for the given function, it's essential to first grasp the general concept. The axis of symmetry is an imaginary vertical line that passes through the vertex of a parabola, dividing it into two symmetrical halves. Think of it as a mirror line; whatever is on one side of the line is mirrored exactly on the other side. This line is critical for understanding the graph and behavior of a quadratic function. The axis of symmetry always has the equation x = h, where 'h' is the x-coordinate of the vertex of the parabola. Identifying this line helps us quickly visualize and analyze quadratic functions, making it a fundamental concept in algebra and calculus. Understanding the axis of symmetry allows us to predict the function's behavior, find its minimum or maximum value, and sketch its graph more accurately. In essence, it serves as a key landmark for deciphering quadratic equations and their corresponding parabolic curves. This concept is not just limited to textbook problems; it appears in real-world applications, such as designing parabolic reflectors, optimizing projectile motion, and even in the architecture of bridges and arches. Therefore, a solid grasp of the axis of symmetry is beneficial for both academic and practical purposes. By finding the axis of symmetry, we can easily determine the vertex, which is the highest or lowest point on the parabola. This is particularly useful in optimization problems, where we need to find the maximum or minimum value of a quadratic function. For example, a business might use this concept to find the price that maximizes profit, or an engineer might use it to design a bridge that can support the most weight. Therefore, understanding the axis of symmetry is not only an important mathematical skill, but also a practical tool that can be applied in many different fields.

Decoding the Function: f(x) = -(x + 9)(x - 21)

The given function, f(x) = -(x + 9)(x - 21), is a quadratic function presented in factored form. Understanding this form is key to finding the axis of symmetry efficiently. The factored form of a quadratic equation is generally expressed as f(x) = a(x - r₁)(x - r₂), where 'a' is a constant and r₁ and r₂ are the roots or x-intercepts of the function. In our case, a = -1, r₁ = -9, and r₂ = 21. These roots are the points where the parabola intersects the x-axis, which are crucial for determining the parabola's symmetry. The coefficient 'a' determines the direction and 'width' of the parabola; if 'a' is negative, the parabola opens downwards, and if 'a' is positive, it opens upwards. In our example, since 'a' is -1, the parabola opens downwards, indicating that the function has a maximum value. The factored form allows us to immediately identify these roots without needing to perform additional calculations such as using the quadratic formula. Knowing the roots makes it straightforward to find the midpoint between them, which is the x-coordinate of the vertex and also the axis of symmetry. This form provides a direct pathway to understanding the symmetrical nature of the quadratic function and its graphical representation. Moreover, the factored form is particularly useful in problem-solving scenarios where the roots are easily identifiable or given, as it simplifies the process of finding the axis of symmetry and other key features of the parabola. This streamlined approach not only saves time but also deepens the understanding of how different forms of quadratic equations reveal specific properties of the function. The factored form also helps in visualizing the graph of the parabola. By knowing the roots and the direction of opening, we can sketch a rough graph of the function, which can be useful in many applications, such as solving inequalities or finding the intervals where the function is positive or negative.

Finding the Axis of Symmetry

The axis of symmetry is the vertical line that passes through the midpoint of the two roots (x-intercepts) of the quadratic function. To find the axis of symmetry for f(x) = -(x + 9)(x - 21), we first identify the roots. From the factored form, we can see that the roots are x = -9 and x = 21. These are the values of x that make the function equal to zero. The next step is to calculate the midpoint between these two roots. The midpoint is simply the average of the two roots, which is calculated by adding the roots together and dividing by 2. In our case, the midpoint is (-9 + 21) / 2. Performing the calculation, we get 12 / 2, which equals 6. Therefore, the x-coordinate of the vertex is 6. Since the axis of symmetry is a vertical line passing through the vertex, its equation is x = 6. This means that the parabola is symmetrical around the vertical line x = 6. Everything on the left side of this line is mirrored on the right side, and vice versa. This understanding is crucial for sketching the graph of the parabola and for understanding the behavior of the function. Knowing the axis of symmetry also helps us find the vertex of the parabola, which is the point where the parabola reaches its maximum or minimum value. In this case, since the coefficient of x² is negative, the parabola opens downwards, and the vertex is the maximum point. To find the y-coordinate of the vertex, we substitute x = 6 into the original function: f(6) = -(6 + 9)(6 - 21) = -15 * -15 = 225. Therefore, the vertex of the parabola is at the point (6, 225). This gives us a complete picture of the parabola: we know its roots, its axis of symmetry, and its vertex. With this information, we can easily sketch the graph of the function and understand its behavior.

Step-by-Step Calculation

Let’s break down the calculation step-by-step to ensure clarity. First, we identify the roots of the function f(x) = -(x + 9)(x - 21). The roots are the values of x that make the function equal to zero. Setting each factor to zero, we have:

• x + 9 = 0, which gives x = -9
• x - 21 = 0, which gives x = 21

So, the roots are -9 and 21. Next, we find the midpoint between these roots. The midpoint formula is (x₁ + x₂) / 2. Plugging in our roots, we get:

Midpoint = (-9 + 21) / 2

Now, let's simplify the expression. First, add -9 and 21:

-9 + 21 = 12

Then, divide the result by 2:

12 / 2 = 6

Thus, the midpoint is 6. This midpoint is the x-coordinate of the vertex of the parabola and also the equation of the axis of symmetry. Therefore, the axis of symmetry is x = 6. To summarize, we identified the roots, calculated the midpoint between the roots, and determined the equation of the axis of symmetry. This step-by-step process provides a clear and concise method for finding the axis of symmetry for any quadratic function in factored form. Each step is logical and builds upon the previous one, making the solution easy to follow. This method is not only useful for this specific function but can be applied to any quadratic function in factored form, making it a valuable tool for solving mathematical problems. By understanding each step, you can confidently find the axis of symmetry for various quadratic functions and gain a deeper understanding of their properties.

The Answer: x = 6

Therefore, after following the steps outlined above, we have definitively determined that the axis of symmetry for the function f(x) = -(x + 9)(x - 21) is x = 6. This means that the parabola represented by this function is perfectly symmetrical around the vertical line x = 6. This result aligns with our understanding of quadratic functions in factored form, where the roots provide direct insight into the axis of symmetry. The roots of the function, -9 and 21, are equidistant from the line x = 6, confirming its role as the axis of symmetry. This symmetry is a fundamental characteristic of parabolas and is crucial for understanding their graphical representation and behavior. Knowing the axis of symmetry allows us to predict the position of the vertex, which is the maximum point in this case since the parabola opens downwards. The axis of symmetry serves as a central reference point for the parabola, making it easier to visualize and analyze its properties. It is not just a numerical answer but a key piece of information that unlocks a deeper understanding of the function's nature. Understanding the concept and calculation of the axis of symmetry is essential for students and anyone working with quadratic functions, providing a powerful tool for solving problems and gaining insights into mathematical relationships. This conclusion reaffirms the importance of understanding the factored form of quadratic functions and how it simplifies the process of finding key characteristics such as the axis of symmetry. The ability to quickly identify and calculate the axis of symmetry is a valuable skill in algebra and calculus, allowing for efficient problem-solving and a deeper appreciation of mathematical concepts.

In conclusion, understanding the axis of symmetry is vital for analyzing quadratic functions. By identifying the roots and calculating their midpoint, we can easily determine the axis of symmetry. For the function f(x) = -(x + 9)(x - 21), the axis of symmetry is x = 6. This knowledge helps us understand the symmetrical nature of the parabola and its graph. For further exploration of quadratic functions and their properties, visit a trusted resource like Khan Academy's Quadratic Functions Section.