Problem 12 - 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. \(x=2 t-1, y=t^{2}+2,\) for \(t\) in \([-10,10]\) window: \([-20,20]\) by \([0,120]\)

Step-by-Step Solution

Answer

The rectangular equation is \(y = \frac{x^2}{4} + \frac{x}{2} + \frac{9}{4}\).

Step 1: Understanding the Problem

We need to graph the parametric equations \(x=2t-1\) and \(y=t^2+2\) for \(t\) in \([-10,10]\) using a graphing calculator. The window for the graph will be set to \([-20,20]\) for the x-axis and \([0,120]\) for the y-axis. After graphing, we will find a rectangular equation that eliminates the parameter \(t\).

Step 2: Using the Graphing Calculator

Set the graphing calculator to parametric mode. Input the equations \(x=2t-1\) and \(y=t^2+2\). Set \(t\) to range from \(-10\) to \(10\). Adjust the window settings to \([-20,20]\) for the x-axis and \([0,120]\) for the y-axis. Graph the equations to visualize the curve.

Step 3: Eliminating the Parameter

To find a rectangular equation (in terms of only \(x\) and \(y\)), we need to eliminate the parameter \(t\) from the parametric equations. Solve the equation for \(x\), \(x = 2t - 1\), to get \(t\) in terms of \(x\): \(t = \frac{x+1}{2}\).

Step 4: Substituting for the Rectangular Equation

Substitute \(t = \frac{x+1}{2}\) into the equation for \(y\), which is \(y = t^2 + 2\). This gives \(y = \left(\frac{x+1}{2}\right)^2 + 2\).

Step 5: Simplifying the Rectangular Equation

Simplify \(y = \left(\frac{x+1}{2}\right)^2 + 2\) to get the final rectangular equation. First, expand the square: \(\left(\frac{x+1}{2}\right)^2 = \frac{(x+1)^2}{4}\), which equals \(\frac{x^2 + 2x + 1}{4}\). Thus, \(y = \frac{x^2 + 2x + 1}{4} + 2\). Finally, simplify this to \(y = \frac{x^2}{4} + \frac{x}{2} + \frac{1}{4} + 2\), or \(y = \frac{x^2}{4} + \frac{x}{2} + \frac{9}{4}\).

Key Concepts

Rectangular Equation

In mathematics, a rectangular equation is one that relates two variables using the Cartesian coordinate system. For parametric equations, you can eliminate the parameter, in this case, the variable \(t\), to form a rectangular equation. This process allows you to express the relationship between \(x\) and \(y\) directly without involving \(t\). Here's how it's done:

• First, you solve one of the parametric equations for \(t\).
• Next, you substitute this expression for \(t\) into the other equation.
• Simplify the resulting equation to express \(y\) purely in terms of \(x\).

For example, in the given problem, you start with \(x = 2t - 1\). Solving for \(t\) gives, \(t = \frac{x+1}{2}\). By substituting this into the \(y\) equation, \(y = t^2 + 2\), you derive the rectangular equation, \(y = \frac{x^2}{4} + \frac{x}{2} + \frac{9}{4}\). This equation now allows us to plot the same curve without depending on \(t\).

Graphing Calculator

A graphing calculator is an essential tool for visualizing mathematical functions, particularly when dealing with parametric equations. It enables you to plot graphs for complex functions that might be difficult to visualize otherwise.To use a graphing calculator for parametric equations like \(x = 2t - 1\) and \(y = t^2 + 2\), follow these steps:

• Switch your calculator to parametric mode. This mode allows you to input equations focusing on a parameter \(t\).
• Enter the given equations and define the range for \(t\), here from \(-10\) to \(10\).
• Adjust the window settings on the calculator to focus on the specific viewing range, such as \([-20, 20]\) horizontally and \([0, 120]\) vertically, as indicated in the problem.
• Finally, generate the graph to view the curve.

Viewing this graph helps you understand the curve's shape and behavior over the specified interval, offering visual insights that complement analytical calculations.

Parameter Elimination

Parameter elimination is a technique where you remove the parameter, \(t\), from parametric equations to find a singular relationship between \(x\) and \(y\). This process provides you with a rectangular equation.Here’s a step-by-step approach to eliminate the parameter:

• Identify the equations connected by the parameter, such as \(x = 2t - 1\) and \(y = t^2 + 2\).
• Solve the simpler equation for \(t\). For instance, from \(x = 2t - 1\), you solve for \(t\) to get \(t = \frac{x + 1}{2}\).
• Substitute this expression for \(t\) into the other equation allowing you to express \(y\) without \(t\).
• Finally, simplify the new equation into a neat rectangular form.

This method not only bridges parametric and rectangular forms but also enhances comprehension of how geometry described parametrically can translate into algebraic form, expanding your analytical toolkit.