Problem 21 - 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. \(x=t+2, y=-\frac{1}{2} \sqrt{9-t^{2}},\) for \(t\) in \([-3,3]\)window: \([-6,6]\) by \([-4,4]\)

Step-by-Step Solution

Answer

The rectangular equation is \(y = -\frac{1}{2}\sqrt{-x^2 + 4x + 5}\).

Step 1: Graph the Parametric Equations

Use the graphing calculator to input the parametric equations:1. Enter the equation for x as \(x = t + 2\).2. Enter the equation for y as \(y = -\frac{1}{2}\sqrt{9-t^2}\).3. Set \(t\) to vary in the window from -3 to 3.4. Set the calculator window to \([-6, 6]\) for x-values and \([-4, 4]\) for y-values. 5. Observe the curve generated by these parametric equations.

Step 2: Eliminate the Parameter

To find a rectangular equation, we need to eliminate the parameter \(t\):1. Start by isolating \(t\) in the equation for \(x\): \[x = t + 2 \implies t = x - 2\]2. Substitute \(t = x - 2\) into the equation for \(y\): \[y = -\frac{1}{2}\sqrt{9 - t^2} = -\frac{1}{2}\sqrt{9 - (x-2)^2}\]

Step 3: Simplify the Rectangular Equation

Now, simplify the equation obtained after eliminating \(t\):1. Expand the expression within the square root: \[9 - (x - 2)^2 = 9 - (x^2 - 4x + 4) = -x^2 + 4x + 5\]2. Substitute back into the equation for \(y\): \[y = -\frac{1}{2}\sqrt{-x^2 + 4x + 5}\]This is the rectangular equation of the curve.

Key Concepts

Graphing Calculator

A graphing calculator is an incredibly helpful tool when working with parametric equations. It allows you to visualize the curve described by the equations, making complex concepts much more tangible. Follow these steps to graph the parametric equations given:

• First, ensure that your calculator is in 'Parametric' mode.
• Enter the equation for the x-component as \(x = t + 2\).
• Next, input the equation for the y-component as \(y = -\frac{1}{2} \sqrt{9-t^2}\).
• Set the parameter \(t\) to range from -3 to 3, aligning with the specified interval.
• Adjust the window settings to display x-values between -6 and 6, and y-values between -4 and 4.

The graphing calculator will now display the trajectory of the parametric equations within the defined window, providing a visual representation of the curve.

Rectangular Equation

Transforming a set of parametric equations into a single rectangular equation involves eliminating the parameter. In this exercise, we are given parametric equations for \(x\) and \(y\) in terms of \(t\). To convert them into a rectangular form, perform the following steps:
First, take the equation for \(x\): \(x = t + 2\). Solve for the parameter \(t\) to get \(t = x - 2\).
Next, substitute \(t\) in the equation for \(y = -\frac{1}{2}\sqrt{9-t^2}\). Replace \(t\) with \(x - 2\) to get:

• \(y = -\frac{1}{2}\sqrt{9 - (x-2)^2}\)

Now you have a rectangular equation that describes the same curve without the parameter \(t\).

Eliminating Parameters

Eliminating parameters is a crucial step when moving from parametric equations to a simpler rectangular equation. The aim is to remove the parameter and express both \(x\) and \(y\) solely in terms of one another. Here's the process clarified:

• First, isolate the parameter \(t\) from one of the equations, preferably the simpler one. Here, from \(x = t + 2\), rearrange to \(t = x - 2\).
• Use this expression for \(t\) and substitute it into the other parametric equation. For this specific case, replace \(t\) in the \(y\) equation: \(y = -\frac{1}{2}\sqrt{9 - t^2}\) becomes \(y = -\frac{1}{2}\sqrt{9 - (x-2)^2}\).

This substitution removes the parameter and lets you derive a more straightforward equation connecting \(x\) and \(y\).

Curve Graphing

Once you have the rectangular equation, the next step is to understand the nature of the curve. By graphing the function, you can study its shape and behavior. Here's how to analyse it:

• Using the rectangular equation \(y = -\frac{1}{2}\sqrt{-x^2 + 4x + 5}\), you can graph by evaluating points or using a graphing utility.
• Notice that the square root limits the domain to real solutions, affecting what parts of the curve are visible.
• Check how the coefficients in the equation influence the steepness, intercepts, and symmetry of the graph.

Understanding curve graphing helps in visual analysis, offering insights into how equations translate into shapes on a coordinate plane.