Problem 13 - Precalculus

Question

(A) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the resulting rectangular equation whose graph represents the curve. Adjust the domain of the rectangular equation, if necessary. $$\begin{aligned} &x=t+2\\\ &y=t^{2} \end{aligned}$$

Step-by-Step Solution

Answer

The curve sketched based on the given parametric equations is an upward parabola which opens upwards and is translated 2 units to the right. The rectangular equivalent of the given parametric equation is \(y = (x - 2)^{2}\) with no restriction on the domain.

Step 1: Understanding the given equations

Having the equations \(x = t + 2\) and \(y = t^{2}\). Two different variables x and y are expressed as functions of a third variable t. Here, t works as a parameter, it describes a certain path along a curve.

Step 2: Sketching the curve

This involves sketching the curve by using the information given in the equations. Start by plotting some points using the parameter t. Let's choose some arbitrary values for t (for example, -2, -1, 0, 1, 2) and calculate the corresponding x and y values. A simple table could be constructed showing corresponding x and y for chosen t values. The orientation of the curve would be from the point corresponding to the lower t value to the point corresponding to the higher t value. Use the points to sketch the curve on the graph.

Step 3: Convert to Rectangular Coordinates

The conversion to rectangular coordinates involves manipulation of the two parametric equations in order to eliminate the parameter t. From the first equation, express t in terms of x as \(t = x - 2\). Substitute this into the second equation \(y = t^{2}\) to eliminate t, giving \(y = (x - 2)^{2}\). This is the final rectangular equation.

Step 4: Adjust the domain if necessary

In this case, there doesn't seem to be any restriction on the values x can take in the rectangular equation. Thus, no adjustment of the domain is necessary.

Key Concepts

Rectangular Coordinates

To start with, let's understand what rectangular coordinates are. These are often referred to as Cartesian coordinates. They are defined in terms of a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on the plane is described by a pair of numbers,

• (x, y), where x is the horizontal distance from the origin
• and y is the vertical distance from the origin.

In the context of the problem, we've transitioned from parametric equations defined by a parameter t to a rectangular coordinate equation.
This involves expressing both x and y as dependent on the same parameter and then finding a relationship solely between x and y.
t is eliminated.>In this exercise, x and y go from being functions of t to being directly related through the equation \(y = (x - 2)^2\). This is a parabolic equation in the rectangular-coordinate system, which we'll explore further in curve sketching.

Curve Sketching

When sketching a curve from parametric equations, the key is to determine the path traced by the parameter.
In our example, the equations were:

• \(x = t + 2\)
• \(y = t^2\).

These equations help us draw a picture of the curve by examining different points as t changes. A useful way to start is by creating a table of values for t, and calculating corresponding x and y values.
Once these points are calculated, you can plot them on a graph. For instance, if you choose t values such as -2, -1, 0, 1, and 2, and compute corresponding points, you'll see how the path evolves.
The curve will have an orientation based on increasing t. In this example, as t increases, the curve moves to the right and upward, reflecting that of a standard parabola opening upwards. This graphical representation of the object defined by our equations helps in visual understanding, giving insights into shapes and directions of curves.

Parameter Elimination

Eliminating the parameter in parametric equations transforms them into their rectangular form.
This is essential for simplifying equations and recognizing familiar shapes. In this exercise, you start with two equations:

• \(x = t + 2\)
• \(y = t^2\)

To eliminate t, rearrange the x equation to express t in terms of x: \(t = x - 2\). Substitute this back into the y equation, \(y = t^2\), to yield \(y = (x - 2)^2\). This process of substitution and manipulation is key to moving from parametric equations to a single variable-defined rectangular equation.
The resulting equation \(y = (x - 2)^2\) is identical to the general form of a parabola. Performing this step simplifies the understanding of the curve, and makes it easier to apply concepts from algebra, such as identifying vertex and axis of symmetry. Navigating from the "parameter world" to coordinates familiar to us can highlight the elegance of mathematical transformations.