Problem 26 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) sketch the curve represented by the parametric equations (indicate the orientation of the curve). Use a graphing utility to confirm your result. (b) Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. $$x=e^{2 t}, y=e^{t}$$

Step-by-Step Solution

Answer

The curve that represents the given parametric equations increases along the x-axis and forms an exponential curve. The equivalent rectangular representation of the curve is \(x = y^{2}\) for \(x > 0\) and \(y > 0\).

Step 1: Understanding the Parametric Equations

The parametric equations given are: \(x = e^{2t}\) and \(y = e^{t}\). It is clear that as \(t\) increases, both \(x\) and \(y\) also increases since the base of the exponential function (\(e\)) is greater than 1. This indicates the orientation of the curve.

Step 2: Sketching the Curve

Plot a few points by subbing different values for \(t\) into \(x = e^{2t}\) and \(y = e^{t}\) and plot those points on a graph to indicate the curve. Also, since \(x\) grows much faster than \(y\) (because of the power of \(t\) is bigger in the \(x\) equation), this means that the curve has the shape of an exponential curve along the x-axis.

Step 3: Eliminating the parameter

In this step, we aim to find an equation in terms of \(x\) and \(y\) only. Observe that \(x = (e^{t})^2\), which is just \(y^{2}\). Hence the required rectangular equation is \(x = y^{2}\).

Step 4: Adjusting the Domain

Given the original parametric equation, both \(x\) and \(y\) are products of an exponential function which means \(x > 0\) and \(y > 0\). Therefore, the required rectangular equation is \(x = y^{2}\) but under the restriction that \(x > 0\) and \(y > 0\).

Key Concepts

Rectangular Equation

A rectangular equation, also known as a Cartesian equation, is an expression that relates two variables, typically denoted as 'x' and 'y,' in a way that can be represented on a standard coordinate system. Unlike parametric equations, which define coordinates using a third variable (the parameter), a rectangular equation gives a direct relationship between 'x' and 'y'.

For the given exercise, the parametric equations are described by an exponential function, and the goal is to eliminate the parameter 't' to form a rectangular equation. By taking advantage of the fact that the variables 'x' and 'y' are both powers of 'e' to the same parameter, we can express 'x' as a function of 'y', leading to the simpler and more familiar rectangular equation of the form: \( x = y^2 \) where 'x' and 'y' are constrained by the conditions 'x > 0' and 'y > 0'. This allows us to understand the behavior of the curve in a conventional x-y coordinate system.

Sketching Curves

Sketching curves is a fundamental skill in mathematics that helps visualize the relationship between two variables described by equations. To sketch a curve effectively, especially for exponential functions, a table of values is often used. Choosing a range of 't' values and calculating the corresponding 'x' and 'y' coordinates provide specific points through which the curve passes.

For our parametric equations \( x = e^{2t} \) and \( y = e^t \) we notice that 'x' increases at a rate that is the square of the rate at which 'y' increases. As a result, plotting these points reveals a curve that grows rapidly along the x-axis—a characteristic of exponential growth. Using graphing technology can also assist in verifying the accuracy of our sketch and understanding the overall shape of the curve.

Eliminating the Parameter

Eliminating the parameter in a set of parametric equations involves finding a single equation that connects the two variables directly. This process is important for converting the parametric representation into a form that's easier to analyze using standard algebraic methods.

In our example, we have \( x = e^{2t} \) and \( y = e^t \) which after observing the relationship between 'x' and 'y', we can express 'x' as a function of 'y' squared. This step simplifies the complex exponential parametric equations into the much simpler \( x = y^2 \) equation. It is crucial to remember, however, that any restrictions on the domain from the original parametric equations should carry over to the rectangular equation.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They have the general form \( f(t) = a^t \) where 'a' is a positive constant. These functions model situations where quantities grow or decay at a rate proportional to their current value.

In this exercise, the base is Euler's number 'e', which is approximately 2.71828, and it's a fundamental base for natural exponential functions. The equations provided, \( x = e^{2t} \) and \( y = e^t \), are exponential functions that describe the rate of growth for 'x' and 'y'. Since 'e' is greater than 1, both 'x' and 'y' increase exponentially, but 'x' does so at a faster rate because of the squared term, distinguishing the nature of the curve.

Graph Orientation

Graph orientation refers to the direction in which a curve moves as the parameter increases. Understanding the orientation is crucial when sketching the path of an object or analyzing movement.

In parametric equations, the orientation can be determined by observing how the 'x' and 'y' values change as the parameter 't' changes. In the provided parametric equations, as 't' increases, both 'x' and 'y' values increase, indicating that the curve travels in a positive direction along both axes. The exponential nature of the functions ensures that the curve will move away from the origin, reflecting the exponential growth as 't' progresses. This insight is essential when plotting the curve to ensure that its direction corresponds with the behavior dictated by the equations.