Equation Of A Parabola With Vertex At (5 3) Explained

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In the realm of quadratic functions, parabolas hold a prominent position, characterized by their distinctive U-shaped curves. The vertex, a crucial point on a parabola, marks either the minimum or maximum value of the function. Understanding the vertex form of a quadratic equation is essential for identifying the vertex coordinates and grasping the parabola's graphical representation. This article delves into the process of determining the equation of a parabola given its vertex, providing a comprehensive explanation and step-by-step guidance.

To effectively identify the equation of a parabola with a specific vertex, it's crucial to understand the vertex form of a quadratic equation. The vertex form provides a direct representation of the parabola's vertex coordinates, making it a valuable tool for analysis and problem-solving. The vertex form of a quadratic equation is expressed as:

y = a(x - h)^2 + k

Where:

  • (h, k) represents the coordinates of the vertex.
  • a determines the direction and steepness of the parabola. If a > 0, the parabola opens upwards, and if a < 0, it opens downwards. The larger the absolute value of a, the steeper the parabola.

In this equation, the vertex coordinates (h, k) are explicitly revealed, making it straightforward to identify the parabola's vertex. The parameter a plays a crucial role in determining the parabola's shape and orientation. A positive value of a indicates that the parabola opens upwards, resembling a U-shape, while a negative value of a signifies that the parabola opens downwards, resembling an inverted U-shape. The magnitude of a also influences the steepness of the parabola; a larger absolute value of a results in a steeper parabola, while a smaller absolute value leads to a flatter parabola.

Understanding the vertex form empowers us to readily extract the vertex coordinates and gain insights into the parabola's graphical characteristics. This knowledge is fundamental for solving problems involving parabolas, such as finding the equation of a parabola given its vertex or determining the vertex from a given equation.

Our primary objective is to pinpoint the equation that represents a parabola with its vertex situated at the coordinates (5, 3). To accomplish this, we will meticulously examine each of the provided options and compare them to the vertex form equation, y = a(x - h)^2 + k. By substituting the given vertex coordinates into the vertex form equation, we can effectively identify the correct equation.

Let's analyze each option:

  • Option A: y = (x - 5)^2 + 3

    Comparing this equation to the vertex form, we observe that h = 5 and k = 3. This aligns perfectly with the given vertex coordinates (5, 3). Therefore, Option A is a potential candidate.

  • Option B: y = (x + 5)^2 + 3

    In this equation, we have h = -5 and k = 3. The x-coordinate of the vertex, h, is -5, which does not match the given vertex (5, 3). Consequently, Option B is not the correct equation.

  • Option C: y = (x - 3)^2 + 5

    Here, h = 3 and k = 5. The vertex coordinates represented by this equation are (3, 5), which differ from the given vertex (5, 3). Thus, Option C is not the equation we seek.

  • Option D: y = (x + 3)^2 + 5

    This equation yields h = -3 and k = 5. The vertex coordinates implied by this equation are (-3, 5), which do not correspond to the given vertex (5, 3). Therefore, Option D is not the correct equation.

By meticulously comparing each option to the vertex form equation and considering the given vertex coordinates, we have successfully identified Option A: y = (x - 5)^2 + 3 as the equation that represents a parabola with its vertex at (5, 3).

Having pinpointed Option A: y = (x - 5)^2 + 3 as the equation representing the parabola with a vertex at (5, 3), let's delve deeper into why this equation aligns with the given vertex coordinates. The vertex form of a quadratic equation, as we've established, is:

y = a(x - h)^2 + k

Where (h, k) denotes the vertex coordinates. In our case, the vertex is (5, 3), so we have h = 5 and k = 3. Substituting these values into the vertex form equation, we get:

y = a(x - 5)^2 + 3

Now, let's examine Option A: y = (x - 5)^2 + 3. This equation perfectly matches the form we derived by substituting the vertex coordinates into the vertex form equation. We can observe that a = 1 in this case, which means the parabola opens upwards. The (x - 5)^2 term ensures that the parabola's vertex lies on the line x = 5, and the + 3 term shifts the parabola vertically upwards by 3 units, placing the vertex at (5, 3).

To further solidify our understanding, let's consider a few points on the parabola. When x = 5, we have:

y = (5 - 5)^2 + 3 = 0 + 3 = 3

This confirms that (5, 3) is indeed a point on the parabola. If we take a point to the left of the vertex, say x = 4, we get:

y = (4 - 5)^2 + 3 = 1 + 3 = 4

And if we take a point to the right of the vertex, say x = 6, we get:

y = (6 - 5)^2 + 3 = 1 + 3 = 4

These calculations demonstrate the symmetry of the parabola around its vertex. The points (4, 4) and (6, 4) are equidistant from the vertex and have the same y-coordinate.

The vertex of a parabola holds immense significance in understanding and graphing quadratic functions. It serves as a pivotal point, providing crucial information about the parabola's shape, position, and direction. The vertex marks the point where the parabola changes direction, either from decreasing to increasing (for parabolas opening upwards) or from increasing to decreasing (for parabolas opening downwards).

When the coefficient a in the vertex form equation y = a(x - h)^2 + k is positive, the parabola opens upwards, and the vertex (h, k) represents the minimum point of the function. Conversely, when a is negative, the parabola opens downwards, and the vertex (h, k) represents the maximum point of the function.

The vertex also plays a vital role in determining the axis of symmetry of the parabola. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is given by x = h, where h is the x-coordinate of the vertex.

Furthermore, the vertex aids in sketching the graph of a parabola. By plotting the vertex and a few additional points, we can accurately depict the parabola's curve. The vertex serves as a reference point, guiding the placement and orientation of the parabola on the coordinate plane.

Parabolas, far from being abstract mathematical concepts, find widespread applications in various fields of science, engineering, and everyday life. Their unique shape and properties make them ideal for designing and analyzing a multitude of systems and structures. Let's explore some compelling examples:

  • Satellite Dishes and Reflectors: The parabolic shape is ingeniously employed in satellite dishes and reflectors to efficiently collect and focus electromagnetic waves, such as radio waves and light. The waves, upon striking the parabolic surface, are reflected towards a single focal point, where a receiver or detector is positioned. This focusing ability enhances the signal strength and sensitivity of these devices.

  • Headlights and Flashlights: The parabolic shape is also utilized in headlights and flashlights to produce a focused beam of light. A light source is placed at the focus of a parabolic reflector, and the light emitted from the source is reflected by the parabolic surface, creating a parallel beam of light that travels in a specific direction. This design ensures that the light is concentrated and directed effectively.

  • Projectile Motion: The trajectory of a projectile, such as a ball thrown into the air or a bullet fired from a gun, closely approximates a parabolic path (ignoring air resistance). This is because the force of gravity acts uniformly on the projectile, causing it to accelerate downwards. The parabolic trajectory allows us to predict the range, maximum height, and time of flight of projectiles.

  • Bridges and Arches: Parabolas are also employed in the design of bridges and arches due to their structural efficiency. The parabolic shape distributes the load evenly, minimizing stress and maximizing stability. Suspension bridges often feature parabolic cables, while arches in bridges and buildings are frequently designed in a parabolic form.

These examples highlight the versatility and practical significance of parabolas in diverse applications. Their ability to focus waves, direct light, model projectile motion, and provide structural support makes them indispensable in various technological and engineering endeavors.

In this comprehensive exploration, we have successfully determined the equation of a parabola with a vertex at (5, 3). By understanding the vertex form of a quadratic equation and meticulously comparing the given options, we identified Option A: y = (x - 5)^2 + 3 as the correct representation. We delved into the significance of the vertex in graphing parabolas and examined the practical applications of parabolas in various fields.

Understanding parabolas is crucial for comprehending quadratic functions and their graphical representations. The vertex form equation provides a powerful tool for analyzing and solving problems involving parabolas. By mastering these concepts, you can confidently tackle a wide range of mathematical challenges and appreciate the practical relevance of parabolas in the world around us.