Problem 27 - Contemporary Precalculus

Question

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. \(x=e^{t}, \quad y=t, \quad\) any real number \(t\)

Step-by-Step Solution

Answer

Question: Eliminate the parameter \(t\) from the given parametric equations \(x = e^t\) and \(y = t\). Answer: \(x = e^y\)

Step 1: Solve for t in the second equation

The second equation is \(y = t\). Solving for \(t\) is straightforward: \(t = y\).

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

We have now found that \(t = y\). Substitute this value of \(t\) into the first equation: \(x = e^t \Rightarrow x = e^y\).

Step 3: Write the final equation

After substituting for \(t\), the equation now only includes \(x\) and \(y\), and we have eliminated the parameter \(t\). The equation is \(x = e^y\). So, the given curve is part of the graph of the equation \(x = e^y\).

Key Concepts

Eliminating Parameters

When dealing with parametric equations, a parameter is often used to represent both variables in a system, typically expressed in terms of one another. This can be very useful in representing curves and other geometries more flexibly. However, sometimes it is necessary to eliminate the parameter to find a more direct relationship between the variables, usually denoted as \(x\) and \(y\).

To eliminate a parameter, you need to express the parameter (in this case, \(t\)) in terms of one of the variables. Here, we had the parametric equations: \(x = e^t\) and \(y = t\). As seen in the solution, you first express \(t\) as `\(t = y\)` using the equation \(y = t\).

• Solve for the parameter in terms of one variable.
• Substitute this expression back into the other equation.

With these steps, you transform the parametric form into a standard form equation, \(x = e^y\), removing the parameter and giving you a clearer direct correlation between \(x\) and \(y\).

Exponential Functions

Exponential functions are a special kind of mathematical function where a constant base is raised to a variable exponent. In the function \(x = e^y\), \(e\) (approximately 2.718) is the base, and \(y\) is the exponent. This kind of function is common in many natural processes, such as compound interest, population growth, and radioactive decay.

Exponential functions have particular properties:

• The base \(e\) is unique and known as Euler's number.
• They exhibit rapid growth or decay, depending on their structure.
• The growth rate increases quickly since it depends on the current value.

When rewriting parametric equations in the form of an exponential function like \(x = e^y\), you can observe how changes in \(y\) result in the exponential increase in \(x\). This demonstrates the transformative nature of exponentiation and its application in crafting equations from real-world scenarios.

Graphing Equations

Graphing equations is an essential skill in understanding the relationship between two variables in a mathematical formula. When graphing the equation \(x = e^y\), you will observe certain notable features:

• The curve always passes through the point (1, 0) since \(e^0 = 1\).
• As \(y\) increases, \(x\) increases exponentially, showcasing rapid growth.
• The graph approaches the \(x\)-axis as \(y\) becomes negative, but never touches or crosses it, reflecting how exponential functions behave.

To graph \(x = e^y\):1. Choose a range for \(y\), such as -3 to 3.2. Calculate \(x\) for each \(y\) value using \(x = e^y\).3. Plot these ordered pairs on a coordinate plane.

Understanding how to graph such equations provides visual insight into their behavior, allowing you to recognize the functional relationships and predict outcomes by visual cues.