Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then find a rectangular equation for the curve. \(x=t+2, y=t^{2},\) for \(t\) in \([-1,1]\)window: \([0,4]\) by \([-2,2]\)

Step-by-Step Solution

Answer

The rectangular equation is \(y = (x - 2)^2\).

Step 1: Understanding the Parametric Equations

The given parametric equations are \(x(t) = t + 2\) and \(y(t) = t^2\). We need to graph these equations using a calculator and convert them into a rectangular equation.

Step 2: Graphing with a Calculator

Input the parametric equations into the graphing calculator: \(x = t + 2\) and \(y = t^2\). Set the parameter \(t\) from \(-1\) to \(1\) and adjust the window settings to \([0, 4]\) for the x-axis and \([-2, 2]\) for the y-axis. Observe the graph generated by the calculator.

Step 3: Isolate t in Terms of x

Start with the equation \(x = t + 2\). To express \(t\) in terms of \(x\), subtract 2 from both sides: \(t = x - 2\).

Step 4: Substitute for y

Use the expression for \(t\) from the previous step, \(t = x - 2\), and substitute it into the equation for \(y\): \(y = t^2\). This gives \(y = (x - 2)^2\).

Step 5: Simplifying the Rectangular Equation

Simplify the rectangular equation obtained: \(y = (x - 2)^2\). This equation represents a parabola that opens upwards and has its vertex at the point (2, 0).

Step 6: Verification and Adjustment to Window

Verify the rectangular equation by plotting \(y = (x - 2)^2\) on the graphing calculator with the same window settings: \([0, 4]\) for \(x\) and \([-2, 2]\) for \(y\). Ensure the plot matches the parameterized curve.

Key Concepts

Rectangular Equations

In mathematics, a rectangular equation is expressed in terms of the standard coordinate system using variables like \(x\) and \(y\). Unlike parametric equations, which use a third variable to express points, rectangular or Cartesian equations provide a direct relationship between \(x\) and \(y\).
For the given parametric equations:

• \(x(t) = t + 2\)
• \(y(t) = t^2\)

To convert these into a rectangular form, we first find \(t\) in terms of \(x\) from the first equation: \(t = x - 2\). Next, substitute this into the second equation: \(y = (x - 2)^2\).
This gives us a rectangular equation representing a parabola. Rectangular equations are crucial for graphing and can be helpful for visualizing relationships in a clear, two-dimensional space.

Graphing Calculator

A graphing calculator is a powerful tool for visualizing and analyzing mathematical functions. It is particularly helpful when dealing with parametric equations. Here's how you can use it:

• Enter each parametric equation separately: \(x = t + 2\) and \(y = t^2\).
• Adjust the graphing window to the desired interval, such as \([0,4]\) for x and \([-2, 2]\) for y.
• Set the parameter \(t\) within a specified range, for example, from \(-1\) to \(1\).

Once the calculator is set, visualize the curve generated by the parametric equations. This visualization helps in understanding how the parameter \(t\) affects the graph of the equations over the selected interval.
Graphing calculators can reinforce the conversion step by visually verifying if the rectangular form \(y = (x - 2)^2\) overlaps with the parametric equation's graph.

Equation Conversion

Equation conversion involves transforming equations from one form to another to simplify or clarify their relationships. In the context of parametric to rectangular forms, you start by expressing one of the parametric variables in terms of \(t\). For instance, with \(x = t + 2\), solve for \(t\) to get \(t = x - 2\).
Next, substitute this expression for \(t\) into the second parametric equation \(y = t^2\), resulting in \(y = (x - 2)^2\).
The outcome is the rectangular equation representing the same curve in a more standard form.
Equation conversion is a critical skill in calculus and analytical geometry. It facilitates understanding, analyzing, and visualizing mathematical concepts. This process helps students bridge the gap between abstract parametric expressions and familiar Cartesian coordinates.