Problem 13 - 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. \(x=2 t, y=t+1,\) for \(t\) in \([-2,3]\)window: \([-8,8]\) by \([-8,8]\)

Step-by-Step Solution

Answer

The rectangular equation is \( y = \frac{x}{2} + 1 \).

Step 1: Understanding the Parametric Equations

The given parametric equations are \( x = 2t \) and \( y = t + 1 \). These describe how the \( x \) and \( y \) coordinates change as the parameter \( t \) varies from \(-2\) to \(3\).

Step 2: Graphing the Parametric Equations

Use a graphing calculator or software to input the parametric equations \( x = 2t \) and \( y = t + 1 \). Set the parameter \( t \) to range from \(-2\) to \(3\), and adjust the window settings to \([-8, 8]\) for both x and y axes. The graph should display a line segment within these bounds.

Step 3: Finding the Rectangular Equation

To find the rectangular equation, express \( t \) in terms of \( x \) using the first equation: \( t = \frac{x}{2} \). Substitute this expression for \( t \) into the second equation: \( y = \frac{x}{2} + 1 \). Hence, the rectangular equation is \( y = \frac{x}{2} + 1 \).

Step 4: Verifying the Solution

Check the rectangular equation \( y = \frac{x}{2} + 1 \) by ensuring it matches the graph obtained from the parametric equations for the given interval of \( t \). The line should be continuous and straight in the graphing calculator's window limits.

Key Concepts

Graphing Calculator

A graphing calculator is a valuable tool when working with parametric equations. It helps visualize the path described by these equations as the parameter changes. To start:

• Input the equations into the calculator: for this example, you'll use the parametric forms given by \( x = 2t \) and \( y = t + 1 \).
• Set the appropriate bounds for \( t \) (from -2 to 3) and adjust the window to view the interval you need (from -8 to 8 for both the x and y axes).
• Observe the graph to understand how the equations map onto a curve or, in this case, a line segment.

With a graphing calculator, you can also explore more insights like points of intersection and symmetry, enhancing your understanding of the parametric path.

Rectangular Equation

Converting parametric equations into a rectangular form can simplify understanding or further application. For the equations \( x = 2t \) and \( y = t + 1 \):

• Express \( t \) in terms of \( x \) using \( x = 2t \), which gives \( t = \frac{x}{2} \).
• Substitute this expression into the equation for \( y \): \( y = \frac{x}{2} + 1 \).

This conversion helps connect parametric equations to more familiar Cartesian coordinates. It provides a straightforward way to graph the relationship as a line \( y = \frac{x}{2} + 1 \), making it easier to analyze or work with using conventional graphing techniques.

Precalculus

In precalculus, students learn fundamental concepts that bridge the gap between algebra and calculus. Parametric equations are a key topic as they describe a path in terms of a parameter, typically denoted as \( t \).

• These equations often offer a more flexible way of defining curves, especially useful for modeling motion and time-dependent scenarios.
• By parameterizing a curve, you can easily express directions that are difficult to model with a simple \( x \) vs. \( y \) equation.

Mastering parametric equations involves practicing with graphing, converting to rectangular forms, and interpreting how changes in \( t \) affect the overall curve. These skills build toward more complex topics in calculus.

Coordinate Geometry

Coordinate geometry uses algebraic equations to describe geometrical shapes. In the case of parametric equations:

• These equations provide a set rule for tracing paths in the coordinate plane as the parameter \( t \) varies.
• Coordinate geometry allows for easy identification of key points, intersections, and characteristics such as slopes and curvature.

Understanding how parametric equations fit into coordinate geometry helps simplify complex curves into understandable components. As seen in the transformation to a rectangular equation, this branch of mathematics makes it easier to manipulate and calculate geometrical properties, preparing you for more advanced applications in math and physics.