Problem 3 - Algebra and Trigonometry

Question

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=t^{2}, \quad y=t-2, \quad 2 \leq t \leq 4 $$

Step-by-Step Solution

Answer

The curve is \( x = (y+2)^2 \), with \( 0 \leq y \leq 2 \).

Step 1: Understand the Parametric Equations

The given parametric equations are \( x = t^2 \) and \( y = t - 2 \) with \( t \) ranging from 2 to 4. These equations describe a curve in terms of the parameter \( t \).

Step 2: Sketch the Curve

To sketch the curve, calculate points for different values of \( t \) within the given range. For instance:- If \( t = 2 \), then \( x = 2^2 = 4 \) and \( y = 2 - 2 = 0 \). So, one point is (4, 0).- If \( t = 3 \), then \( x = 3^2 = 9 \) and \( y = 3 - 2 = 1 \). Another point is (9, 1).- If \( t = 4 \), then \( x = 4^2 = 16 \) and \( y = 4 - 2 = 2 \). The last point is (16, 2).Plot these points on a graph and connect them smoothly to visualize the curve.

Step 3: Eliminate the Parameter

To find a rectangular-coordinate equation, eliminate \( t \) from the parametric equations. Start with the second equation: \( y = t - 2 \). Solve for \( t \):\[ t = y + 2 \]Substitute \( t = y + 2 \) into the first equation \( x = t^2 \):\[ x = (y + 2)^2 \]This is the rectangular equation representing the curve.

Step 4: Define the Domain of the Rectangular Equation

Determine the domain of \( y \) using the given range of \( t \) from 2 to 4. Since \( y = t - 2 \), when \( t = 2 \), \( y = 0 \). When \( t = 4 \), \( y = 2 \). Thus, the domain of \( y \) is \( 0 \leq y \leq 2 \).

Key Concepts

Rectangular-Coordinate Equation

The rectangular-coordinate equation is a way of expressing a curve without involving any parameters, only using the usual coordinates of a graph. This means you'll have an equation formed in terms of \( x \) and \( y \) rather than a third parameter like \( t \). In our example, to find this equation, we started by rearranging one of the parametric equations to solve for \( t \). From \( y = t - 2 \), we derived \( t = y + 2 \). Once we have \( t \) in terms of \( y \), we substitute back into the other equation: \( x = t^2 \), resulting in \( x = (y + 2)^2 \). This equation now describes the original curve using only \( x \) and \( y \), making it a rectangular-coordinate equation. It's important because it simplifies the visualization and analysis of curves by removing the additional parameter.

Curve Sketching

Curve sketching is a fundamental step to understand the nature and behavior of a parametric equation and its equivalent rectangular form. It involves plotting points by substituting values of the parameter \( t \) into the parametric equations, helping visualize the actual shape of the path described by the equations.In our exercise, values for \( t \) range from 2 to 4. By plugging those values into \( x = t^2 \) and \( y = t - 2 \), we generated points:

• \( t = 2 \): Point \( (4, 0) \)
• \( t = 3 \): Point \( (9, 1) \)
• \( t = 4 \): Point \( (16, 2) \)

Plot these points on a coordinate plane and smoothly connect them. The points form a parabolic curve, demonstrating how the curve evolves as \( t \) changes. This visualization makes it easier to relate the mathematical equations to the geometric shape.

Eliminating the Parameter

Eliminating the parameter in parametric equations involves getting rid of \( t \) to express the relationship directly between \( x \) and \( y \). This process simplifies the representation and makes it easier to understand the characteristics of the curve. Starting from the parametric equations \( x = t^2 \) and \( y = t - 2 \), we need to express one in terms of the other. First, solve \( y = t - 2 \) for \( t \), giving \( t = y + 2 \). Substitute this expression in \( x = t^2 \), producing \( x = (y + 2)^2 \).The removal of the parameter not only simplifies analysis but is often required when parametric representations are transformed into forms suitable for specific applications like curve fitting or integration. However, essential characteristics such as domain and range must still be considered.

Domain of a Function

Determining the domain of a function is critical to understanding where the function is defined and valid. When dealing with parametric equations, the parameter range directly influences the domain of the resulting rectangular-coordinate equation.For the given equations, the parameter \( t \) is between 2 and 4. From \( y = t - 2 \), as \( t \) varies, \( y \) ranges from 0 (when \( t = 2 \)) to 2 (when \( t = 4 \)). Therefore, the domain of the curve in terms of \( y \) is \( 0 \leq y \leq 2 \).Understanding the domain lets you know the possible values \( y \) can take in the rectangular-coordinate equation \( x = (y + 2)^2 \), ensuring accurate representation and use of the equation over the intended range. This is crucial in applying the equation to real-world contexts where only certain values are appropriate or possible.