Problem 22 - 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. \(x=t, y=\sqrt{4-t^{2}},\) for \(t\) in \([-2,2]\)window: \([-6,6]\) by \([-4,4]\)

Step-by-Step Solution

Answer

Rectangular equation: \(y = \sqrt{4 - x^2}\). This represents the top half of a circle with radius 2.

Step 1: Understand the Parametric Equations

We are given the parametric equations \(x = t\) and \(y = \sqrt{4 - t^2}\) with \(t\) ranging from \(-2\) to \(2\). We need to graph these equations in the specified window: \([-6,6]\) by \([-4,4]\).

Step 2: Graph the Parametric Equations

Use a graphing calculator to input the parametric equations. Set the interval for \(t\) from \(-2\) to \(2\). The graph should be plotted in the x-window from \(-6\) to \(6\) and y-window from \(-4\) to \(4\). You should observe a semi-circular curve as the result.

Step 3: Derive Rectangular Equation from Parametric Equations

To find the rectangular equation, we use the relationship between \(x\) and \(t\) given by \(x = t\). Substitute \(t = x\) into the equation for \(y\): \(y = \sqrt{4 - x^2}\). This substitution gives the rectangular form of the equation.

Step 4: Interpret the Rectangular Equation

The derived rectangular equation, \(y = \sqrt{4 - x^2}\), represents the top half of a circle with radius 2, centered at the origin. This is consistent with the observation from the graph, a semi-circle spanning from \(-2\) to \(2\) on the x-axis.

Key Concepts

Graphing Calculator

Graphing calculators are powerful tools that simplify visualizing mathematical expressions. They allow users to input equations and view corresponding graphs, which can help in understanding complex relationships between variables effortlessly. To visualize parametric equations like the ones in our example, it's important to set the parameters correctly in the calculator.

When dealing with parametric equations, ensure to:

• Input the formulas for both x and y using the parameter, in this case, it's the variable t.
• Set the parameter range, here from -2 to 2, to get the desired section of the curve.
• Adjust the window settings to view the appropriate portion of the graph, specifically the x-window from -6 to 6 and y-window from -4 to 4.

Interacting with a graphing calculator in this manner can help you quickly see the semi-circular shape produced by the parametric equations. It illustrates the practical application of technology in solving math problems.

Rectangular Equation

In mathematics, a rectangular equation is derived from parametric equations by eliminating the parameter. This transformation results in an equation with just two variables, typically x and y, which is easier to work with for many analyses.

For the equations provided, we have parametric forms: \ x = t \ and \ y = \sqrt{4 - t^2}\. To convert these into a rectangular form, solve for the parameter t and substitute it back into the other equation. Since x equals t, you can replace t with x in the equation for y:

• Start with: \(y = \sqrt{4 - t^2}\)
• Substitute: \(t = x\)
• Result: \(y = \sqrt{4 - x^2}\)

This new rectangular equation \(y = \sqrt{4 - x^2}\) describes the same curve but without the parameter t. This form is advantageous as it clearly shows the relationship between x and y directly, which can be particularly useful in graphing or algebraic manipulation.

Semi-Circle

A semi-circle is half of a circle, formed by cutting a whole circle into two equal parts through its diameter. When working with equations, a semi-circle is represented by a specific form.In our problem, the equation \(y = \sqrt{4 - x^2}\) represents the upper half of a circle because:

• The square root function produces only non-negative values.
• The expression \(4 - x^2\) confines the shape to a bounded area since \(x^2\) cannot exceed 4, meaning x must be between -2 and 2, as a square of a real number is always non-negative.
• The constant 4 under the square root indicates that the radius is 2, as a circle is the set of points at a fixed distance (the radius) from a central point.

Thus, this equation graphs as a semi-circle with a radius of 2 centered at the origin, spanning from \ x = -2 \ to \ x = 2 \ on the x-axis. This serves as an elegant example of how parametric and rectangular equations can be used to model familiar geometric shapes.