Rectangular Form Of Parametric Equations X=6t And Y=t^2/4

In the realm of mathematics, parametric equations offer a powerful way to describe curves and relationships between variables. Parametric equations define variables, such as x and y, in terms of a third independent variable, often denoted as t, which acts as a parameter. Understanding how to convert parametric equations into rectangular form is crucial for visualizing and analyzing the relationships they represent. This article delves into the process of transforming the given parametric equations, x = 6t and y = t²/4, into their equivalent rectangular form, providing a step-by-step guide and illuminating the underlying concepts.

Understanding Parametric and Rectangular Equations

Before we dive into the conversion process, let's establish a solid understanding of the two types of equations we're dealing with. Parametric equations, as mentioned earlier, express variables x and y as functions of a parameter t. This parameter allows us to trace the path of a curve as t varies. In contrast, rectangular equations define a direct relationship between x and y, typically in the form y = f(x) or x = g(y). Rectangular equations provide a static representation of the curve, without explicitly involving a parameter.

The advantage of parametric equations lies in their ability to describe complex curves and motions that are difficult or impossible to represent with a single rectangular equation. For example, the motion of a projectile, which involves both horizontal and vertical displacement changing over time, can be elegantly described using parametric equations. Similarly, curves with self-intersections or loops are often more easily defined parametrically.

However, rectangular equations offer a more direct way to visualize the relationship between x and y. They allow us to quickly identify the shape of the curve and its key features, such as intercepts, asymptotes, and symmetry. The ability to convert between parametric and rectangular forms provides us with the flexibility to choose the representation that best suits our needs for analysis and visualization. Converting from parametric to rectangular form involves eliminating the parameter t to obtain a direct relationship between x and y. This often involves algebraic manipulation and substitution, as we'll demonstrate in the following steps.

Step-by-Step Conversion Process

Now, let's embark on the journey of converting the given parametric equations, x = 6t and y = t²/4, into their rectangular form. The fundamental principle behind this conversion is to eliminate the parameter t, thereby establishing a direct link between x and y.

Step 1: Solve for t in one of the equations

The first step involves isolating the parameter t in one of the given equations. Looking at our equations, x = 6t seems like the easier equation to manipulate. To solve for t, we simply divide both sides of the equation by 6:

x / 6 = (6t) / 6

This simplifies to:

t = x / 6

We have now successfully expressed t in terms of x. This expression will be crucial in the next step.

Step 2: Substitute the expression for t into the other equation

Next, we substitute the expression for t that we obtained in Step 1 into the other parametric equation, which is y = t²/4. Replacing t with x/6, we get:

y = (x/6)² / 4

This substitution effectively eliminates the parameter t from the equation, leaving us with a relationship between x and y.

Step 3: Simplify the equation

The final step involves simplifying the equation obtained in Step 2 to arrive at the rectangular form. Let's start by expanding the square:

y = (x² / 36) / 4

Now, to divide by 4, we can multiply by its reciprocal, which is 1/4:

y = (x² / 36) * (1/4)

Multiplying the fractions, we get:

y = x² / 144

Therefore, the rectangular form of the given parametric equations is y = x²/144. This equation represents a parabola opening upwards, with its vertex at the origin.

Analyzing the Rectangular Equation

Having successfully converted the parametric equations to rectangular form, y = x²/144, we can now analyze the resulting equation to gain insights into the curve it represents. The equation y = x²/144 is a quadratic equation, which is a hallmark of a parabola. The standard form of a parabola opening upwards or downwards is y = a(x - h)² + k, where (h, k) is the vertex of the parabola and a determines the parabola's shape and direction. In our case, we can rewrite the equation as y = (1/144)x², which corresponds to the standard form with a = 1/144, h = 0, and k = 0.

This tells us that the parabola opens upwards (since a is positive) and its vertex is located at the origin (0, 0). The value of a, which is 1/144, dictates how wide or narrow the parabola is. A smaller value of a results in a wider parabola, while a larger value results in a narrower parabola. In this case, the parabola is relatively wide due to the small value of a. To further understand the parabola, we can find its intercepts. The y-intercept is the point where the parabola intersects the y-axis, which occurs when x = 0. Substituting x = 0 into the equation, we get y = 0²/144 = 0. Thus, the y-intercept is at the origin (0, 0), which we already knew was the vertex. The x-intercepts are the points where the parabola intersects the x-axis, which occur when y = 0. Setting y = 0 in the equation, we get 0 = x²/144. This implies that x² = 0, so x = 0. Therefore, the only x-intercept is also at the origin (0, 0). This confirms that the vertex is indeed the lowest point on the parabola. By analyzing the rectangular equation, we have gained a comprehensive understanding of the curve represented by the original parametric equations. We know it's a parabola opening upwards, with its vertex at the origin, and we have a good sense of its shape and intercepts.

Choosing the Correct Answer

Having successfully derived the rectangular form of the parametric equations, y = x²/144, we can now confidently identify the correct answer from the given options. The options provided are:

A. y = x²/144 B. y = x²/9 C. y = x/144 D. y = x/9

Comparing our derived rectangular equation, y = x²/144, with the options, it is clear that option A, y = x²/144, is the correct answer. The other options represent different types of curves. Option B, y = x²/9, is also a parabola opening upwards, but it is narrower than the one we derived. Options C and D, y = x/144 and y = x/9, represent straight lines, not parabolas. Therefore, by carefully following the conversion process and simplifying the resulting equation, we have accurately determined the rectangular form and selected the correct answer from the given choices. This exercise demonstrates the importance of understanding the relationship between parametric and rectangular equations and the ability to convert between them.

Conclusion

In conclusion, we have successfully converted the parametric equations x = 6t and y = t²/4 into their rectangular form, which is y = x²/144. This process involved solving for the parameter t in one equation, substituting that expression into the other equation, and simplifying the result. The rectangular equation reveals that the curve represented by the parametric equations is a parabola opening upwards, with its vertex at the origin. This conversion highlights the power of parametric equations in describing curves and the importance of being able to translate them into rectangular form for analysis and visualization. The ability to work with both parametric and rectangular representations provides a comprehensive toolkit for understanding and manipulating mathematical relationships. By mastering these techniques, we can tackle a wider range of problems and gain deeper insights into the world of mathematics.