Problem 15 - 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. \(x=t+2, y=-\frac{1}{2} \sqrt{9-t^{2}},\) for \(t\) in \([-3,3]\) window: \([-6,6]\) by \([-4,4]\)

Step-by-Step Solution

Answer

Rectangular equation: \( y = -\frac{1}{2} \sqrt{5 - x^2 + 4x} \).

Step 1: Plotting the Curve

Use a graphing calculator to plot the parametric equations \( x = t + 2 \) and \( y = -\frac{1}{2} \sqrt{9 - t^2} \) for \( t \) in the interval \([-3, 3]\). Set the viewing window from \([-6, 6]\) on the x-axis and \([-4, 4]\) on the y-axis. Observe the shape and behavior of the curve.

Step 2: Expressing in Rectangular Form

To convert the parametric equations to a rectangular form, start by isolating \( t \) from the equation \( x = t + 2 \) which gives \( t = x - 2 \). Substitute this expression for \( t \) in the second equation \( y = -\frac{1}{2} \sqrt{9 - t^2} \). This gives \( y = -\frac{1}{2} \sqrt{9 - (x - 2)^2} \).

Step 3: Simplifying the Equation

The expression \( 9 - (x - 2)^2 \) can be expanded to simplify. First, note \( (x - 2)^2 = x^2 - 4x + 4 \). Substitute into the equation to get \( y = -\frac{1}{2} \sqrt{5 - x^2 + 4x} \). Now, further simplify if possible, but this is the rectangular form of the equation.

Key Concepts

Parametric Equations

Parametric equations allow us to express a curve by defining both the x and y coordinates in terms of a third variable, usually denoted as \(t\). This approach is incredibly useful for graphing complex curves that may be difficult to express as a standard function. It essentially provides a set of instructions on how to draw the curve by plotting individual points for various values of \(t\).

• In our exercise, \(x = t + 2\) and \(y = -\frac{1}{2} \sqrt{9 - t^2}\) are the parametric equations given for the plane curve.
• They help in outlining the path that one would plot to obtain the curve span from \(t = -3\) to \(t = 3\).

Using these equations, we can explore different intervals of \(t\) to visualize how a curve is formed, which can offer insights into its geometrical properties such as symmetry or intersections.

Graphing Calculator

A graphing calculator serves as an incredibly powerful tool to visualize parametric equations. It facilitates the plotting process by calculating the corresponding \(x\) and \(y\) coordinates for every value of \(t\) within the specified interval, creating a more intuitive understanding of the curve's behavior.

• Setting up the calculator involves entering the parametric forms, specifying the range for \(t\), and adjusting the window size for both \(x\) and \(y\) axes, like in our exercise with a range of \([-6, 6]\) for \(x\) and \([-4, 4]\) for \(y\).
• This setup helps in magnifying particular aspects of the curve for closer inspection or simplifies it to gain a broad perspective.

With the graphing calculator, students can dynamically interact with mathematical concepts, which enhances their comprehension and retention of the material.

Interval for Parameter t

Choosing an interval for the parameter \(t\) is a crucial aspect in parametric plotting since it determines the section of the graph that will be shown. The interval may affect whether parts of a function or certain behaviors are visible in a graph.

• In our case, the interval \([-3,3]\) was chosen for \(t\). This choice directly influences the endpoints of the curve.
• Including these endpoints ensures the curve depicts a complete picture rather than a partial or misleading one, capturing the full designated path of a parametric equation.

Carefully selecting and adjusting the interval enables more control over which parts of a graph are displayed and analyzed, which can be crucial for understanding or solving mathematical problems.

Curve Plotting

Curve plotting with parametric equations is like connecting dots with a smooth line. For each value of parameter \(t\) within your specified interval, corresponding points \((x, y)\) are plotted to illustrate the curve.

• The detailed step-by-step solution showcases how to generate these points and visualize the curve's path.
• It emphasizes the importance of correctly setting up a viewing window, in this case from \([-6, 6]\) by \([-4, 4]\). This window is key to capturing the entirety of the curve efficiently.

Understanding the visual aspect of curve plotting enriches one's ability to comprehend complex behaviors and characteristics of curves beyond mere equations. Through practice, students can develop the skill to mentally visualize these plots before activating their graphing calculators, giving them a stronger grasp of the subject.