Problem 3 - 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. $$x=t^{2}, \quad y=t-2, \quad 2 \leq t \leq 4$$

Step-by-Step Solution

Answer

Curve: Plot points (4,0), (9,1), (16,2). Rectangular equation: \( x=(y+2)^2 \).

Step 1: Understand Parametric Equations

The given parametric equations are \( x = t^2 \) and \( y = t - 2 \), with the parameter \( t \) varying from 2 to 4. These equations describe a curve in the xy-plane by expressing the coordinates \( x \) and \( y \) as functions of the parameter \( t \).

Step 2: Sketch the Curve

To sketch the curve, compute \( x \) and \( y \) for values of \( t \) ranging from 2 to 4. \( t = 2 \) gives \( (x, y) = (4, 0) \), \( t = 3 \) gives \( (9, 1) \), and \( t = 4 \) gives \( (16, 2) \). Plot these points in the xy-plane and join them smoothly to visualize the curve.

Step 3: Eliminate the Parameter

We need to find a rectangular-coordinate equation (in terms of \( x \) and \( y \)) by eliminating \( t \). Start by solving the equation \( y = t - 2 \) for \( t \), giving \( t = y + 2 \). Substitute this into the equation for \( x = t^2 \): \( x = (y + 2)^2 \).

Step 4: Express the Resulting Equation

The resulting equation in rectangular coordinates is \( x = (y + 2)^2 \), which describes the curve in the xy-plane without a parameter. This can be rewritten as \( y = \pm \sqrt{x} - 2 \) if needed for further analysis.

Step 5: Verify the Range

Check if the values cover the entire specified range of \( t \). Since \( 2 \leq t \leq 4 \), \( x = t^2 \) will vary from 4 to 16, and \( y = t - 2 \) will vary from 0 to 2. This confirms that our rectangular equation correctly represents the original parametric range.

Key Concepts

Eliminating the Parameter

When working with parametric equations, one of the common tasks is to eliminate the parameter. This process allows you to express the curve using a single equation in the Cartesian coordinate system, making it easier to analyze and graph. In our exercise, we have the parametric equations:

• \( x = t^2 \)
• \( y = t - 2 \)

To eliminate the parameter \( t \), follow these steps:

1. Start with the equation for \( y \): \( y = t - 2 \).
2. Solve for \( t \): \( t = y + 2 \).
3. Substitute this expression into the equation for \( x \): \( x = (y + 2)^2 \).

Now, \( x = (y + 2)^2 \) is a rectangular-coordinate equation representing the curve without the parameter \( t \). This equation is particularly useful because it retains the relationship between \( x \) and \( y \) in a simpler form.

Rectangular-Coordinate Equation

The transition from parametric equations to a rectangular-coordinate equation provides a different perspective of the curve. The rectangular-coordinate equation for this exercise, \( x = (y + 2)^2 \), is not only easier to plot but also reveals more about the geometric nature of the curve.This particular form indicates that the curve is a parabola. Here, the expression \((y + 2)^2\) suggests a parabola that opens horizontally. In mathematical terms, the variable \( y \) is the symmetrical axis of the parabola, and it opens in the positive \( x \) direction. Additionally, this equation helps to identify key features, such as:

• Vertex of the parabola: At the point where \( y = -2 \), \( x \) becomes zero. Hence, the vertex is at (0, -2).
• The curve's range: Since \( t \) ranges from 2 to 4, \( y \) spans from 0 to 2, affirming the curve stretching within these limits in the xy-plane.

Understanding the rectangular form allows for a clearer interpretation of how the curve behaves and where it lies in the coordinate system.

Visualizing Curves

Visualizing curves starts with plotting the parametric points in the xy-plane. Given the parametric equations in this exercise:

• \( x = t^2 \)
• \( y = t - 2 \)

We calculate sets of coordinates for various values of \( t \) within its defined range of 2 to 4. First, plug \( t \) into both equations:

• When \( t = 2 \), the point is \((4, 0)\).
• When \( t = 3 \), the point is \((9, 1)\).
• When \( t = 4 \), the point is \((16, 2)\).

Plotting these points provides a framework for drawing the curve. By smoothly connecting these plotted points, we visually trace the trajectory of the parabola in the xy-plane. Visualizing this curve thus becomes an intuitive way to interpret and understand the underlying geometric shape represented by the equations. Such sketches help in connecting the algebraic form with its graphical representation, bridging the gap between abstract math and geometric intuition.