Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a set of parametric equations to represent the graph of the given rectangular equation using the parameters (a) \(t=x\) and (b) \(t=2-x.\) $$y=\ln (x+2)$$

Step-by-Step Solution

Answer

The set of parametric equations with \(t=x\) is \(x = t\) and \(y = \ln(t+2)\). The set of parametric equations with \(t=2-x\) is \(x = 2-t\) and \(y = \ln(4-t)\).

Step 1: Substitute \(t=x\) to find parametric equation

By using the parameter \(t=x\), we get the equation \(y=\ln (t+2)\). The set of parametric equations with \(t=x\) is therefore \(x = t\) and \(y = \ln(t+2)\).

Step 2: Solve \(t=2-x\) for x

Before substituting \(t\) as \(2-x\), we need to solve for \(x\) in terms of \(t\). Here \(x = 2-t\).

Step 3: Substitute \(x = 2-t\) in \(y = \ln(x+2)\)

Now if we substitute \(x = 2-t\) into the original equation \(y = \ln(x+2)\), we obtain \(y = \ln((2-t) + 2)\), which simplifies to \(y = \ln(4-t)\).

Key Concepts

Rectangular Equations

Rectangular equations provide us with a familiar way of describing curves on a Cartesian plane. They are equations in the form of functions where we express one variable explicitly in terms of another, typically written as \(y = f(x)\).

The equation \(y = \ln(x+2)\) is an example of a rectangular equation. In this context, 'rectangular' simply means the equation is written in terms of \(x\) and \(y\), coordinates that we often plot on traditional graph paper.

The function involves a natural logarithm, which you can recognize by the symbol \(\ln\). Understanding rectangular equations is crucial for converting them into parametric forms as shown in the provided exercise.

Parameterization

Parameterization is the process of expressing a mathematical object using parameters, which in many cases simplifies the understanding and representation of curves or surfaces.

In the problem, the original rectangular equation \(y = \ln(x+2)\) is expressed using parameters \(t=x\) and \(t=2-x\).

This means:

• For \(t = x\), we substitute \(x\) for \(t\) directly, resulting in the parametric equations: \(x = t\) and \(y = \ln(t+2)\).
• For \(t = 2-x\), we solve for \(x\) as a function of \(t\) (\(x = 2-t\)), and substitute into the rectangular equation: \(y = \ln(4-t)\).

This transformation allows us to plot the same curve using different information, which can be more intuitive for complex shapes or movements.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a logarithm with a special base: the number \(e\), an irrational constant approximately equal to 2.71828.

The natural logarithm has interesting properties that make it important in mathematics, especially in calculus and complex analysis.

• The function \(\ln(x)\) increases slowly as \(x\) increases.
• It is undefined for \(x \leq 0\). This is important to remember when working with logarithmic functions.
• \(\ln(1) = 0\) because \(e^0 = 1\).

In our exercise, we deal with \(y = \ln(x+2)\), which means we have translated the logarithm horizontally 2 units to the left. This allows the function to be defined for \(x > -2\). Recognizing how these transformations affect the domain of a function is critical when finding parametric forms.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate a desired variable or to simplify a mathematical expression.

It is an essential skill when transforming rectangular equations into parametric equations.

In the original exercise:

• When given \(t=2-x\), the goal was to solve for \(x\) in terms of \(t\) before substituting back into the original function, yielding \(x = 2-t\).
• This required straightforward algebraic steps: rearranging the equation to isolate \(x\).

Algebraic manipulation can become more complex for higher-level mathematics, but the fundamental techniques remain invaluable for solving and simplifying equations. In this exercise, it helps bridge between rectangular and parametric representations, illuminating different aspects of the same function.