Problem 21 - 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. $$x=2 t, y=|t-2|$$

Step-by-Step Solution

Answer

The curve has two separate lines, depends on the value of \(t\). For \(t \geq 2\), the line is \(y = \frac{x}{2} - 2\); and for \(t < 2\), the line is \(y = 2 - \frac{x}{2}\). These lines intersect at the point (4, 0). Therefore, there are no adjustments necessary for the domain.

Step 1: Sketch the Parametric Curve

Use the parametric equations \(x = 2t\) and \(y = |t - 2|\) to generate a set of points for \(t\). It's helpful to separate the function \(y = |t - 2|\) into two functions, \(y = t - 2\) for \(t \geq 2\) and \(y = 2 - t\) for \(t < 2\). Now, generate a set of points for each case, sketch these points on a graph, and indicate the orientation.

Step 2: Use a Graphing utility to confirm

Confirm the sketch by using a graphing utility. Enter the equations \(x = 2t\) and \(y = |t - 2|\), then confirm the shape and orientation matches the sketch from step 1.

Step 3: Eliminate the parameter

To eliminate the parameter, express \(t\) in terms of \(x\) from the first equation. Thus, \(t = \frac{x}{2}\). Substitute \(t\) in the second equation, yielding \(y = |\frac{x}{2} - 2|\).

Step 4: Adjust the domain if necessary

Since there are no restrictions on \(x\), the rectangular equation is valid for all values. Hence, no adjustment for the domain is necessary.

Key Concepts

Rectangular Equations

When dealing with parametric equations, we often want to find an equivalent rectangular equation. This means we aim to eliminate the parameter from the set of equations to obtain one that relates the variables directly. In our exercise, given the parametric equations \(x = 2t\) and \(y = |t - 2|\), we can convert these into a rectangular equation. To do this, express the parameter \(t\) as a function of \(x\): \(t = \frac{x}{2}\). Substitute this expression into the second equation to get \(y = |\frac{x}{2} - 2|\). This is the rectangular form of the given parametric equations.This form allows us to understand the graph of the function without considering both \(x\) and \(y\) as functions of another variable (\(t\)). Essentially, it's the way we typically represent mathematical functions.

Domain Adjustment

After transforming parametric equations to rectangular equations, we need to consider domain adjustments. This involves determining the appropriate range of \(x\) values that are meaningful for the rectangular equation. By doing this, we ensure that the domain accurately reflects the behavior observed in the parametric format.In our specific case, since the original parametric equations do not impose restrictions on the values that \(x\) can take (as all real numbers are valid for \(t\)), the domain for the rectangular equation \(y = |\frac{x}{2} - 2|\) remains all real numbers. This means that there's no need to further adjust the domain from what's apparent initially.Remember, domain adjustments are essential when restrictions are either apparent in the parametric equations or result from the transformations themselves.

Graphing Utility

Using a graphing utility is an essential step in visualizing parametric equations and verifying the resulting rectangular equations. These tools allow us to input and graph complex equations quickly, offering a clear interface for checking our work. In our exercise, once we identified our equations as \(x = 2t\) and \(y = |t - 2|\), we used a graphing utility to confirm this parametric path visually. By entering the equations, the graphing tool showed us the trajectory of the path, its direction, and ensured that our initial hand sketch was accurate. Additionally, it provided a method to see that the rectangular equation \(y = |\frac{x}{2} - 2|\) produces the same visual graph, which reconfirms its correctness. Graphing utilities are an invaluable resource for corroborating mathematical results and enhancing comprehension through visual aids. They help to confirm the shapes and orientations of curves derived from both parametric and rectangular equations effectively.