Problem 10 - A Graphical Approach to College Algebra

Question

For each plane curve, 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. \(x=t^{2}, y=\sqrt{t},\) for \(t\) in \([0,4]\) window: \([-2,20]\) by \([0,4]\)

Step-by-Step Solution

Answer

The rectangular equation for the curve is \( x = y^4 \).

Step 1: Plot the Parametric Equations

First, let's input the given parametric equations into a graphing calculator. Enter \( x = t^2 \) and \( y = \sqrt{t} \) and set the parameter \( t \) to range from 0 to 4.

Step 2: Set the Graphing Window

The graphing window specified is \([-2, 20]\) for the x-values and \([0, 4]\) for the y-values. Adjust these settings in your calculator so you can observe the full graph of the curve within the specified window.

Step 3: Observe and Analyze the Curve

Once graphed, observe the shape and behavior of the curve on the screen to understand how the parameter \( t \) affects the position of \(x\) and \(y\). This will help in determining the rectangular equation.

Step 4: Eliminate the Parameter

To convert the parametric equations into a rectangular equation, eliminate the parameter \( t \). From \( y = \sqrt{t} \), express \( t \) in terms of \( y \): \( t = y^2 \). Substitute \( t = y^2 \) into the equation for \( x \): \( x = (y^2)^2 = y^4 \).

Step 5: Verify the Rectangular Equation

Check the rectangular equation \( x = y^4 \) by comparing the curve created by this equation over the specified window to the one plotted in Step 1. The curves should match, confirming the correctness of the rectangular form.

Key Concepts

Parametric Equations

Parametric equations represent a powerful way to express the coordinates of points on a curve in terms of a separate parameter, usually denoted as \( t \). These equations define both \( x \) and \( y \) as functions of \( t \). For instance, in the given equations \( x = t^2 \) and \( y = \sqrt{t} \), both coordinates depend on the value of \( t \).
Parametric equations are often used to describe complex curves that aren't easily represented by traditional rectangular (Cartesian) equations. They are especially useful in physics for modeling paths and motion, where time or another independent variable serves as the parameter. This approach can make it easier to understand how each component of a coordinate changes over time or another factor. While potentially more complex initially, they offer a more flexible approach to tackling multi-dimensional problems.

Graphing Calculator

Graphing calculators are essential tools for visualizing mathematical concepts like parametric equations. They allow us to input these equations and examine the resulting graphs over a given range. To graph parametric equations, enter each one separately in the calculator, and adjust settings to suit the range of the parameter \( t \) and the viewable window.
A graphing calculator makes it easy to tweak these settings, showing the immediate effects of changing variables or the window. This helps in understanding how different elements of an equation interact and affect the graph. In our example, setting \( t \) from 0 to 4 and adjusting the graphing window to \([-2, 20]\) for \(x\) and \([0, 4]\) for \(y\) allows us to see the entire curve without truncation. Observing the graph helps in verifying that the eventual rectangular form accurately represents the parametric equations.

Parameter Elimination

Eliminating the parameter \( t \) is a fundamental step in converting parametric equations to a rectangular form, which describes the relationship between \( x \) and \( y \) without involving \( t \). This process often requires rearranging one of the equations to express \( t \) in terms of one of the coordinates.
In the given problem, we start with \( y = \sqrt{t} \), which can be rewritten as \( t = y^2 \). This expression for \( t \) is then substituted into the equation for \( x \), resulting in \( x = (y^2)^2 = y^4 \). Thus, \( x = y^4 \) is the rectangular equation. This elimination confirms that the relationship between \( x \) and \( y \) can be expressed without the parameter, simplifying the understanding of the curve's structure. It's crucial to verify that this new equation matches the original curve produced by the parametric forms, ensuring no features were lost during conversion.