Problem 8 - Precalculus with Limits

Question

In Exercises 7-26, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (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. \(x=3-2t\) \(y=2+3t\)

Step-by-Step Solution

Answer

The graph of the parametric equations is a line running from the upper right to the lower left, as the x-coordinates decrease and the y-coordinates increase with increasing value of 't'. The corresponding rectangular equation is \(y = 5.5 - 1.5x\), which has the full range of real numbers as the domain.

Step 1: Plot the parametric equations

Begin with sketching the parametric equations. One common approach is to create a table of values for different 't' values. For example, we could start with t=0 and then progress to t=1, t=2 etc. Substituting these values into the parametric equations will yield corresponding x and y coordinates. These coordinates dictate the points on the graph.

Step 2: Indicate the orientation

The orientation of the curve is determined by the direction in which t increases. In this case, as 't' increases, 'x' decreases and 'y' increases. This means that the curve will start from a higher x-value and run to a lower x-value while going from a lower y-value to a higher y-value, thus moving from right to left while going up. On the graph, indicate the direction using arrows.

Step 3: Eliminate the parameter

To convert these parametric equations into a rectangular equation, we need to eliminate the parameter 't'. For this, solve one of the parametric equations for 't', i.e. rewrite one of the equations to make 't' the subject. For instance, from the equation \(x = 3-2t\), we can express 't' as \(t = (3-x)/2.\) Then, substitute these expressions into the other parametric equations to eliminate ‘t’.

Step 4: Finalize the Rectangular Equation

Substituting 't' from the previous step into the equation for 'y' gives \(y = 2 + 3*((3-x)/2) = 2+ (9 - 3*x)/2 = 2 + 9/2 - 3x/2 = 5.5 - 1.5x\). So the corresponding rectangular equation is \(y = 5.5 - 1.5x\).

Step 5: Adjust the Domain

Since 't' can be any real number in the parametric equations, 'x' and 'y' span the full range of real numbers in the rectangular equation, so there's no restriction on the domain.

Key Concepts

Rectangular Equations

When working with parametric equations, one common task is converting them into rectangular equations. This process involves removing the parameter, often 't', to express the relationship between 'x' and 'y' directly. In the given exercise, we start with the parametric equations:

• \(x = 3 - 2t\)
• \(y = 2 + 3t\)

To find the rectangular equation, we solve one of the equations for 't'. From the equation \(x = 3 - 2t\), we can find 't' as \(t = \frac{3-x}{2}\). Next, we substitute this expression for 't' into the equation for 'y':\(y = 2 + 3\left(\frac{3-x}{2}\right)\).Simplifying, we get:\(y = 5.5 - 1.5x\).This rectangular equation, \(y = 5.5 - 1.5x\), represents the same curve as the parametric equations, but in a more conventional 'x-y' form.

Curve Sketching

Sketching the curve of a parametric equation involves plotting points for various values of 't' and drawing the trajectory. For the equations:

• \(x = 3 - 2t\)
• \(y = 2 + 3t\)

select different 't' values such as 0, 1, 2, etc. This method helps create a table of points:When \(t = 0\), \(x = 3\) and \(y = 2\).When \(t = 1\), \(x = 1\) and \(y = 5\).When \(t = 2\), \(x = -1\) and \(y = 8\).Use these points to sketch the graph. As 't' increases, the curve starts at point (3, 2) and moves towards (-1, 8). This visual representation makes it easier to understand how 'x' and 'y' change concerning 't'.

Eliminating Parameter

Eliminating the parameter 't' from parametric equations is a crucial skill for converting them into the rectangular form. This step allows us to see the direct relationship between 'x' and 'y'. For the parametric pair:

• \(x = 3 - 2t\)
• \(y = 2 + 3t\)

we solve \(x = 3 - 2t\) for 't', getting \(t = \frac{3-x}{2}\). Substitute this into \(y = 2 + 3t\), and we derive the equation \(y = 5.5 - 1.5x\).This process eliminates 't' and provides a simple linear equation. Make sure to check your math at each step for accuracy. By expressing the relationship in terms of 'x' and 'y', we simplify analysis and graphing.

Graph Orientation

Graph orientation is essential to understand the direction in which a curve is drawn when plotted. It reveals how the curve behaves as 't' increases. For:

• \(x = 3 - 2t\)
• \(y = 2 + 3t\)

As 't' increases, \('x'\) decreases while \('y'\) increases. This means the curve starts from the rightmost high point and descends leftward as it ascends vertically.Indicating this orientation on the graph is crucial.Draw arrows along the plotted curve in the direction 't' advances, here from right to left. These arrows assist in quickly comprehending the path traced by the parametric equations over time. This insight can be particularly beneficial when determining how the curve interacts with other graphs or axes.