Problem 16 - 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. $$x=3-2 t, y=2+3 t$$

Step-by-Step Solution

Answer

The curve represented by the parametric equations \(x=3-2t, y=2+3t\) is a straight line with negative slope. After eliminating the parameter, we get the rectangular equation \(y=-3/2x+9/2\).

Step 1: Sketch the curve

The curve can be sketched by plotting a few points. For example, when \(t=0\), we get the point \(x=3\) and \(y=2\). When \(t=1\), we get the point \(x=1\) and \(y=5\) and so on. By plotting these points and connecting them, we can get a sense of the curve. Remember to also indicate the orientation of the curve by drawing arrows showing that the curve is traced out as \(t\) increases.

Step 2: Confirm your curve with a graphing utility

Using a graphing utility that accepts parametric equations such as Desmos or a graphing calculator, input the equations \(x=3-2t, y=2+3t\) and verify that the plot matches the sketched curve from step one.

Step 3: Eliminate the parameter

To eliminate the parameter \(t\) from the equations, first solve the equation \(x=3-2t\) for \(t\) to get \(t=(3-x)/2\). Substitute this into the second equation \(y=2+3t\) to get \(y=2+3((3-x)/2)\), which simplifies to \(y=-3/2x+9/2\).

Step 4: Adjust the domain if necessary

In this case, no adjustment of the domain of the resulting rectangular equation is necessary because the original parameter \(t\) can take any values.

Key Concepts

Rectangular Equation

To move from a parametric equation to a more traditional form, we often eliminate the parameter, typically noted as "t". This conversion results in what is called a rectangular equation. For the given parametric equations, we have:

• \(x = 3 - 2t\)
• \(y = 2 + 3t\)

Our goal is to eliminate \(t\). To do this, solve one of the equations for \(t\) and substitute into the other. For instance, solve \(x = 3 - 2t\) for \(t\):

• \(t = \frac{3 - x}{2}\)

Now, substitute \(t\) into \(y = 2 + 3t\):

• \(y = 2 + 3\left( \frac{3-x}{2} \right)\)
• This simplifies to \(y = -\frac{3}{2}x + \frac{9}{2}\)

This equation \(y = -\frac{3}{2}x + \frac{9}{2}\) is the rectangular equation, where the relationship between \(x\) and \(y\) doesn't depend explicitly on \(t\). In our specific case, no domain adjustments were necessary.

Graphing Utility

Graphing utilities, such as graphing calculators or online tools like Desmos, play an essential role in visualizing mathematical concepts. They allow for confirmation of hand-drawn graphs and provide an interactive way to explore changes in equations.These tools support input for parametric equations—important for verifying sketches obtained manually. Here’s how you can use a graphing utility:

• Enter the parametric equations: \(x = 3 - 2t\) and \(y = 2 + 3t\).
• Set the parameter \(t\) over the interval you are interested in exploring.
• Observe the graph, ensuring that it matches any sketch you initially created manually.

Not only do graphing utilities help in checking work, but they also offer insight into the nature of the curve, such as symmetry, intercepts, and general direction. For students, these technologies make abstract mathematical concepts more tangible.

Curve Orientation

Understanding curve orientation is crucial when dealing with parametric equations. When visualizing these curves, the orientation gives you insight into how the curve is traced as the parameter "t" increases.When sketching manually, start by plotting points for various values of \(t\):

• Example: For \(t = 0\), we find point \((3, 2)\).
• For \(t = 1\), the point is \((1, 5)\).

Connect these points in order, and you will start to see a path or trajectory forming. The direction in which this path is traced is the orientation of the curve. On your sketch, show this by adding arrows:

• Draw arrows from point-to-point to indicate the path's direction as \(t\) increases.

In utilizing a graphing utility, the orientation can often be visually confirmed, reaffirming the direction you inferred manually. Understanding this orientation helps not only in sketching but also in solving real-world problems where movement and direction matter.