Problem 15 - 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. $$\begin{aligned} &x=\sqrt{t}, y=3 t-4, \text { for } t \text { in }[0,4]\\\ &\text { window: }\lfloor- 6,6\rfloor \text { by }\lfloor- 6,10\rfloor \end{aligned}$$

Step-by-Step Solution

Answer

The rectangular equation is \(y = 3x^2 - 4\).

Step 1: Identify Parameter Equations

We have the parameter equations: \(x = \sqrt{t}\) and \(y = 3t - 4\) for \(t\) in the interval [0,4]. Our task is to convert these into a rectangular equation and graph them in the given window.

Step 2: Explore the Range and Domain for the Graph

For \(x = \sqrt{t}\), as \(t\) ranges from 0 to 4, \(x\) ranges from 0 to 2. For \(y = 3t - 4\), when \(t = 0\), \(y = -4\); when \(t = 4\), \(y = 8\). Thus, the graph will be plotted for \(x\) values from 0 to 2 and \(y\) values from -4 to 8 inside the specified window \([-6,6] \text{ by } [-6,10]\).

Step 3: Eliminate the Parameter

To eliminate \(t\), solve the equation \(x = \sqrt{t}\) for \(t\). We get \(t = x^2\). Substitute this expression for \(t\) into the equation for \(y\): \(y = 3(x^2) - 4 = 3x^2 - 4\). This gives the rectangular equation \(y = 3x^2 - 4\).

Step 4: Graph the Rectangular Equation

Using a graphing calculator, plot the equation \(y = 3x^2 - 4\) within the specified window \([-6,6]\) for \(x\) and \([-6,10]\) for \(y\). The curve appears as a parabola opening upwards, starting at \(y = -4\) when \(x = 0\) and reaching \(y = 8\) when \(x = 2\).

Step 5: Validate Graph in Window's Range

Ensure that the curve correctly represents the original parametric equations within the given intervals. Compare the graph limits with \(x\) from 0 to 2 and \(y\) from -4 to 8 to verify accuracy.

Key Concepts

Rectangular Equations

Rectangular equations are expressions that use the standard Cartesian coordinate system with variables typically denoted as \(x\) and \(y\). These equations describe the relationship between \(x\) and \(y\) without involving an additional parameter like \(t\). In most cases, rectangular equations are easier to visualize on a standard graph, as they define a direct relationship between the x and y axes. This can be helpful for quickly identifying key graphical features, such as intercepts, asymptotes, or symmetry.

• It directly relates \(x\) and \(y\) in a two-dimensional plane.
• With rectangular equations, you don't need an intermediate variable to find one variable based on another.
• Many common equations, like those for circles, ellipses, and parabolas, are naturally rectangular.

Developing a rectangular equation from a parametric form can make complex curves more intuitive and straightforward to analyze. This form of equation eliminates the necessity for calculating or guessing values of \(t\), as used in parametric equations.

Graphing Calculator

Graphing calculators are powerful tools for visualizing complex mathematical equations, including both parametric and rectangular equations. They allow you to graph functions over specified intervals and within a user-defined window. These calculators can perform numerical calculations quickly and provide almost instantaneous visual feedback of mathematical relationships. When working with parametric equations like \(x = \sqrt{t}\) and \(y = 3t - 4\), setting up your graphing calculator correctly is essential for accurate representation.

• Input each parametric equation into the calculator's parametric mode.
• Set the viewing window to encompass the ranges specified, such as \([-6,6]\) for \(x\) and \([-6,10]\) for \(y\).
• Check the plot by comparing calculated points with expected outcomes from your equation calculations.

Modern calculators often feature options to toggle between different forms of equations and can automatically convert parametric forms into rectangular form, helping to streamline your analysis and verify solutions.

Converting Parametric to Rectangular Form

Converting parametric equations into rectangular form involves eliminating the parameter \(t\) to achieve a single equation relating \(x\) and \(y\). This is an essential skill for simplifying complex curves and making them easier to graph and understand without a parameter value. For example, with the equations \(x = \sqrt{t}\) and \(y = 3t - 4\), solving for \(t\) in terms of \(x\) and substituting this back into the \(y\) equation allowed us to find a single relationship:

• First, solve one equation for \(t\), e.g., \(t = x^2\) from \(x = \sqrt{t}\).
• Substitute \(t = x^2\) into the other equation: \(y = 3t - 4\) becomes \(y = 3(x^2) - 4\).
• Simplify as needed for the final rectangular form, here, \(y = 3x^2 - 4\).

This process is crucial for those needing a direct \(x-y\) relationship for use in further analysis or graphical representation. Whatever values you choose for \(x\), you can immediately calculate corresponding \(y\) values, yielding a more traditional graph.