Problem 9 - 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. \(x=\sqrt{t}, y=3 t-4,\) for \(t\) in \([0,4]\) window: \([-6,6]\) by \([-6,10]\)

Step-by-Step Solution

Answer

The rectangular equation is \( y = 3x^2 - 4 \).

Step 1: Analyze Parametric Equations

The given parametric equations are \( x = \sqrt{t} \) and \( y = 3t - 4 \). The parameter \( t \) varies from 0 to 4, and we need to explore how these values of \( t \) define the curve.

Step 2: Express Parameter in Terms of x

To find a rectangular equation, express \( t \) in terms of \( x \) using one of the parametric equations. From \( x = \sqrt{t} \), we can solve for \( t \) by squaring both sides: \( t = x^2 \).

Step 3: Substitute to Find Rectangular Equation

Substitute \( t = x^2 \) into the equation for \( y \): \( y = 3t - 4 \). This gives \( y = 3(x^2) - 4 \). Therefore, the rectangular equation is \( y = 3x^2 - 4 \).

Step 4: Interpret the Rectangular Equation

The rectangular equation \( y = 3x^2 - 4 \) represents a parabola that opens upwards with vertex at (0, -4). This matches the characteristics of the parametric curve over the specified window and parameter interval.

Key Concepts

Rectangular Equations

Rectangular equations, also known as Cartesian equations, express relationships between variables, primarily using the coordinates \(x\) and \(y\). In the context of parametric equations, you often start with equations that involve a third variable \(t\), known as the parameter.
In our example, we started with the parametric equations \(x = \sqrt{t}\) and \(y = 3t - 4\). To convert these into a rectangular equation, you solve for the parameter \(t\) in terms of one of the coordinate variables, here \(x\). This yielded \(t = x^2\). Substituting \(t\) into the equation for \(y\) gives you the rectangular equation \(y = 3x^2 - 4\).
This resulting equation describes the same relationship between \(x\) and \(y\) in a more familiar format, without explicitly involving the parameter \(t\). Understanding how to transition from parametric to rectangular equations is beneficial when analyzing mathematical models as it simplifies computations and graphing in many cases.

Graphing Calculator

A graphing calculator is a powerful tool that allows you to visualize mathematical equations and functions. It is especially useful when working with plane curves and parametric equations.

• Graphing calculators let you input complex equations and view their graphs almost instantly.
• They provide a visual aid to verify the behavior and shape of equations like the one obtained in our exercise: \(y = 3x^2 - 4\).
• Setting the correct window according to the problem is crucial. In our case, we used a window of \([-6,6] \times [-6,10]\), ensuring the curve fits properly within the view.

Utilizing a graphing calculator offers a better understanding of how different mathematical functions behave by allowing for multiple trials and visual comparisons.

Plane Curve

A plane curve is essentially a curve that lies on a two-dimensional plane. When dealing with parametric equations, such as \(x = \sqrt{t} \) and \(y = 3t - 4\), they define a set of points forming a curve on this plane.
The curve's behavior is determined by varying the parameter \(t\), which in our problem was restricted to the interval \([0, 4]\). As \(t\) changes, the pair \((x,y)\) traces out the shape of the curve, providing insight into its geometric nature.
In this scenario, converting to a rectangular equation \(y = 3x^2 - 4\) describes a well-known shape: a parabola. Understanding plane curves is fundamental for visualizing mathematical relationships and exploring different mathematical topics including polynomials, trigonometry, and calculus.