Problem 76 - Precalculus

Question

Find a set of parametric equations to represent the graph of the rectangular equation using (a) \(t=x\) and \((b) t=2-x\) $$y=e^{2 x}$$

Step-by-Step Solution

Answer

Parametric equations for (a) are \(x=t\), \(y=e^{2t}\) and for (b) are \(x=2-t\), \(y=e^{4-2t}\).

Step 1: Parametric Equation for (a) \(t=x\)

Here, we are given that \(t=x\). Substituting \(x\), the rectangular equation becomes: \(y=e^{2t}\). This gives us a set of parametric equations: \(x=t\) and \(y=e^{2t}\).

Step 2: Parametric Equation for (b) \(t=2-x\)

In this case, we are given that \(t=2-x\). To substitute \(x\) in the rectangular equation, first solve for \(x\) in terms of \(t\), which gives us \(x=2-t\). Substituting \(x\) in the rectangular equation, we get: \(y=e^{2(2-t)} = e^{4-2t}\). Thus, the parametric equations are: \(x=2-t\) and \(y=e^{4-2t}\).

Key Concepts

Rectangular Equation

A rectangular equation is a mathematical expression that relates two variables, usually denoted as \( x \) and \( y \), in a standard Cartesian coordinate system. The equation in this exercise is \( y = e^{2x} \), which is a classic example of a rectangular equation involving an exponential component. Rectangular equations serve as the basis for defining curves and plots on a two-dimensional plane.

• These equations are directly plotted on the Cartesian axis.
• They do not use parameters, just the variables \( x \) and \( y \).
• They help in visualizing the relationship between the variables.

Transforming a rectangular equation into a parametric form can provide a more detailed view of the relationship between variables and is helpful for various computational applications or when expressing complex motion or relationships.

Exponential Functions

Exponential functions are a type of mathematical function in which an independent variable appears in the exponent. In our rectangular equation, the exponential function is \( y = e^{2x} \). Exponential functions are characterized by a constant base raised to a variable exponent, and they show rapid growth or decay depending on the power.

• They are often used to model natural phenomena such as population growth, radioactive decay, and interest calculations.
• The base \( e \) is a mathematical constant, approximately equal to 2.718, known as Euler's number.
• An exponential function defined by \( y = e^{ax} \) is called a 'natural exponential function' if the base is \( e \).

In this exercise, the function \( y = e^{2x} \) grows exponentially at a rate proportional to itself, demonstrating a powerful increase as \( x \) becomes larger.

Coordinate Transformation

Coordinate transformation involves changing the position of a set of coordinates from one system to another. This process can change how we view and interpret equations graphically. A transformation might include translating, scaling, or rotating the coordinates within their plane. In this exercise, we explore a parametric transformation with \( t = x \) and \( t = 2-x \).

• For \( t = x \), the transformation simplifies to \( x = t \) and \( y = e^{2t} \).
• For \( t = 2-x \), it results in \( x = 2-t \) and \( y = e^{4-2t} \).

These parametric transformations effectively change the original view of the rectangular equation, providing different insights and highlights different features of the curve. They are particularly useful in describing looping and reversing movements which might not be obvious in standard rectangular form.