Quadratic Function: Vertex, Focal Length, And Focal Point

Understanding quadratic functions is a cornerstone of algebra and calculus. This article dives into the specifics of a given quadratic function to identify key features such as its vertex, focal length, and focal point. We will walk through the process step-by-step, ensuring a clear grasp of the underlying concepts.

Determining the Vertex (h, k) of a Quadratic Function

When we talk about the vertex of a quadratic function, we're referring to the point where the parabola changes direction. It's either the highest point (maximum) or the lowest point (minimum) on the graph. For a quadratic function in the form y = ax² + bx + c, the vertex form is y = a(x - h)² + k, where (h, k) represents the coordinates of the vertex. This form is incredibly useful because it directly reveals the vertex, making it easy to visualize and analyze the parabola.

Now, let's apply this to our given function: y = -1/16 x² + 1. Notice that this equation is already in a simplified form that resembles the vertex form. We can rewrite it as y = -1/16 (x - 0)² + 1. By comparing this to the general vertex form, y = a(x - h)² + k, we can easily identify the values of h and k. In this case, h = 0 and k = 1. Therefore, the vertex of the quadratic function y = -1/16 x² + 1 is located at the point (0, 1). This means the parabola's highest point (since the coefficient of x² is negative) is at (0, 1). Understanding the vertex is crucial because it gives us a central reference point for the parabola, helping us to understand its symmetry and overall shape. This point serves as a cornerstone for graphing and further analysis of the function. The vertex not only tells us the maximum or minimum value of the function but also provides insight into the axis of symmetry, which is the vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.

Calculating the Focal Length (c')

The focal length, denoted as c', is another critical parameter of a parabola. It's the distance between the vertex and the focus, as well as the distance between the vertex and the directrix. The focus is a fixed point on the interior of the parabola, while the directrix is a fixed line on the exterior. These elements play a key role in the parabola's definition: every point on the parabola is equidistant from the focus and the directrix. The focal length helps determine the 'width' or 'steepness' of the parabola; a smaller focal length results in a narrower parabola, while a larger focal length creates a wider one.

To find the focal length c', we use the relationship between the coefficient of the x² term (which we'll call a) and c'. The formula that connects these is c' = 1 / (4|a|). This formula tells us that the focal length is inversely proportional to the absolute value of a. In our equation, y = -1/16 x² + 1, the coefficient a is -1/16. Now, let's plug this value into the formula:

c' = 1 / (4|-1/16|) c' = 1 / (4 * 1/16) c' = 1 / (1/4) c' = 4

So, the focal length c' for the parabola y = -1/16 x² + 1 is 4 units. This value is essential for understanding the parabola's curvature and its relationship to the focus and directrix. A focal length of 4 indicates that the focus is 4 units away from the vertex, which will be crucial when we determine the coordinates of the focal point. The focal length is not just a numerical value; it's a geometric property that helps us visualize and understand the shape of the parabola. By knowing the focal length, we can accurately sketch the parabola and identify its key characteristics.

Locating the Focal Point (0, p)

The focal point is a specific point within the parabola that, along with the directrix, defines the curve's shape. For a parabola that opens upwards or downwards, like the one in our example, the focal point lies on the axis of symmetry. Since our parabola y = -1/16 x² + 1 opens downwards (due to the negative coefficient of the x² term), the focal point will be located below the vertex.

We know the vertex is at (0, 1) and the focal length c' is 4. To find the y-coordinate of the focal point (which we'll call p), we need to consider the direction in which the parabola opens. Because it opens downwards, we subtract the focal length from the y-coordinate of the vertex:

p = k - c' p = 1 - 4 p = -3

Therefore, the focal point is located at (0, -3). This point is a crucial element of the parabola, as it dictates how the curve reflects rays. If we were to shine rays parallel to the axis of symmetry onto the parabola, they would all converge at the focal point. This property has significant applications in optics and antenna design. The focal point, along with the vertex and focal length, provides a complete picture of the parabola's geometry. By understanding these elements, we can accurately describe and analyze the behavior of quadratic functions.

Conclusion

In summary, for the quadratic function y = -1/16 x² + 1, we have determined the following:

• Vertex (h, k): (0, 1)
• Focal Length (c'): 4
• Focal Point (0, p): (0, -3)

These parameters provide a comprehensive understanding of the parabola's shape and position. By mastering these concepts, you can confidently analyze and interpret quadratic functions in various mathematical and real-world contexts. Further exploration of parabolas and their properties can lead to a deeper appreciation of their role in mathematics and its applications.

For additional resources and further reading on quadratic functions, consider visiting Khan Academy's Algebra Section. This trusted website offers comprehensive lessons and practice exercises to enhance your understanding.