Problem 4 - Precalculus: Mathematical for Calculus

Question

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=2 t+1, \quad y=\left(t+\frac{1}{2}\right)^{2}$$

Step-by-Step Solution

Answer

Sketch a parabola from the parametric points, rectangular equation is \( y = \frac{x^2}{4} \).

Step 1: Understand the Parametric Equations

We have the parametric equations given as \( x = 2t + 1 \) and \( y = \left(t + \frac{1}{2}\right)^2 \). These equations define \( x \) and \( y \) in terms of a third variable \( t \), which is the parameter.

Step 2: Sketch the Curve

To sketch the curve, create a table of values for \( t \), calculate corresponding \( x \) and \( y \) values, and plot these points in the coordinate plane. Let \( t \) take values like -1, 0, 1, 2 and so on. For each \( t \), calculate \( x = 2t + 1 \) and \( y = \left(t + \frac{1}{2}\right)^2 \). Plot the points (x, y): - For \( t = 0 \), \( x = 1, y = \frac{1}{4} \)- For \( t = 1 \), \( x = 3, y = \frac{9}{4} \)- For \( t = -1 \), \( x = -1, y = \frac{1}{4} \)- And so on.This helps you identify the shape of the curve.

Step 3: Express \( t \) in terms of \( x \)

From the equation \( x = 2t + 1 \), solve for \( t \):\[ t = \frac{x - 1}{2} \]

Step 4: Substitute \( t \) in \( y \) Equation

Replace \( t \) in the \( y \) equation with \( \frac{x - 1}{2} \). This gives:\[ y = \left(\frac{x - 1}{2} + \frac{1}{2}\right)^2 = \left(\frac{x}{2}\right)^2 \]

Step 5: Simplify the Rectangular Equation

Simplify the expression obtained in the last step:\[ y = \left(\frac{x}{2}\right)^2 = \frac{x^2}{4} \]This is the rectangular equation of the curve.

Key Concepts

Rectangular Coordinates

In mathematics, rectangular coordinates, often referred to as Cartesian coordinates, provide a way to describe a point in a plane using a pair of numbers that specify its horizontal and vertical positions. These coordinates are typically written as \((x, y)\), where:

• \(x\) represents the horizontal position on the plane.
• \(y\) represents the vertical position.

To convert parametric equations to rectangular coordinates, we eliminate the parameter (in this case, \( t \)) to express the relationship between \( x \) and \( y \) directly. This process results in a single equation that defines the curve in the coordinate plane. For example, in our problem, the parametric equations \( x = 2t + 1 \) and \( y = \left(t + \frac{1}{2}\right)^2 \) can be transformed into the rectangular equation \( y = \frac{x^2}{4} \). This equation allows us to understand and analyze the curve’s behavior using a more familiar \((x, y)\) coordinate system.

Curve Sketching

Curve sketching is a critical skill in graphically understanding the shape and behavior of equations. It involves plotting points on a graph to visualize the relationship between variables. To sketch curves from parametric equations, we follow these steps:

• Select a series of values for the parameter \( t \), both positive and negative, to capture the entire curve.
• Calculate the corresponding \( x \) and \( y \) values using the parametric equations.
• Plot these \((x, y)\) points on a coordinate plane.

Following this process for the given parametric equations by choosing values such as \( t = -1, 0, 1 \), we compute the points \((-1, \frac{1}{4}), (1, \frac{1}{4}), (3, \frac{9}{4})\) and so on. Plotting these points gives a more concrete understanding of the curve’s shape, which is a parabola in this case. By connecting the points smoothly, we achieve a complete sketch of the curve, capturing its essential features.

Eliminating Parameters

Eliminating parameters is a technique to convert parametric equations into a single rectangular equation. This involves expressing one parameter in terms of another variable, usually \( x \) or \( y \). In our exercise, we followed these steps:

• First, solve for the parameter \( t \) using one of the parametric equations, such as \( x = 2t + 1 \). Rearrange to find \( t = \frac{x - 1}{2} \).
• Substitute this expression into the other equation \( y = \left(t + \frac{1}{2}\right)^2 \).
• Simplify the resulting expression to derive the rectangular equation \( y = \frac{x^2}{4} \).

This process simplifies the analysis of the curve by removing the dependency on \( t \) and expressing the curve solely in terms of \( x \) and \( y \). Eliminating parameters not only makes the curves easier to sketch but also allows for a deeper exploration of their properties, such as symmetry, intercepts, and vertex points.