Problem 1 - 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} $$ x=2 t, \quad y=t+6 $$

Step-by-Step Solution

Answer

The curve is a straight line: \( y = \frac{x}{2} + 6 \).

Step 1: Understand the Parametric Equations

The given parametric equations are \( x = 2t \) and \( y = t + 6 \). Here, \( t \) is the parameter, and as \( t \) changes, the values of \( x \) and \( y \) change, tracing out a path or curve.

Step 2: Sketch the Curve

Begin by selecting various values for \( t \) and calculate corresponding \( x \) and \( y \) values.- If \( t = 0 \), then \( x = 2(0) = 0 \) and \( y = 0 + 6 = 6 \).- If \( t = 1 \), then \( x = 2(1) = 2 \) and \( y = 1 + 6 = 7 \).- If \( t = -1 \), then \( x = 2(-1) = -2 \) and \( y = -1 + 6 = 5 \).Plot these points (and a few more if necessary) on a graph and draw a line through them. You should see a straight line.

Step 3: Eliminate the Parameter

Eliminate \( t \) from the equations to find a cartesian equation. Solve \( x = 2t \) for \( t \): \[ t = \frac{x}{2} \] Substitute into \( y = t + 6 \):\[ y = \frac{x}{2} + 6 \]Thus, the rectangular-coordinate equation is \( y = \frac{x}{2} + 6 \).

Step 4: Verify the Rectangular Equation

Check if the derived equation \( y = \frac{x}{2} + 6 \) is consistent with the parametric equations by substituting values from the parametric table calculated in Step 2. You should find that for each point calculated, the rectangular equation holds true, confirming the line equation is correct.

Key Concepts

Rectangular-Coordinate Equation

Parametric equations are often used because they provide a way of describing a curve using parameters like time or a specific variable. In our original exercise, the given equations are:

• \(x = 2t\)
• \(y = t + 6\)

Here, the parameter \(t\) gives us pairs of \(x\) and \(y\) coordinates. When we want to remove this parameter to get a direct relationship between \(x\) and \(y\), we transition to a rectangular-coordinate equation (also known as a Cartesian equation).
To eliminate the parameter \(t\), solve \(x = 2t\) for \(t\), resulting in \(t = \frac{x}{2}\). Substituting this into the equation \(y = t + 6\) gives: - \(y = \frac{x}{2} + 6\)
This is our rectangular-coordinate equation, providing a direct link between \(x\) and \(y\) without the parameter. The parameter's elimination simplifies the relationship and focuses on how these coordinates behave together.

Eliminating the Parameter

When working with parametric equations, the process of eliminating the parameter involves transforming the equations to a more familiar and direct form. This is crucial for understanding the overall shape and behavior of the curve in question.
To eliminate the parameter in our example, we start with the two given equations: \(x = 2t\) and \(y = t + 6\). Our goal is to express \(y\) solely in terms of \(x\), eliminating \(t\) altogether.
The steps involved to eliminate the parameter are:

• Solve one of the parametric equations for the parameter. In this case, \(t\) is isolated from \(x = 2t\) giving \(t = \frac{x}{2}\).
• Substitute this expression for \(t\) into the second parametric equation, \(y = t + 6\). This leads to the equation \(y = \frac{x}{2} + 6\).

Through this method, you can now see the curve's equation without needing to refer back to the parameter, providing a clear and concise relationship between \(x\) and \(y\). It is a transformative process that helps in understanding and visualizing mathematical relationships more clearly.

Curve Sketching

Curve sketching is an essential topic in mathematics to interpret and visualize the behavior of functions or equations. With parametric equations, the process involves tracing out the path that the points (\(x, y\)) will follow as the parameter \(t\) changes.
To sketch the curve from the given parametric equations \(x = 2t\) and \(y = t + 6\), you begin by choosing several values for the parameter \(t\) and then calculate the corresponding \(x\) and \(y\) values. For instance:

• When \(t = 0\), \(x = 0\) and \(y = 6\).
• When \(t = 1\), \(x = 2\) and \(y = 7\).
• When \(t = -1\), \(x = -2\) and \(y = 5\).

By plotting these points and others on a coordinate plane, you notice that they align to form a straight line. This linearity indicates that the path traced by the parametric equations is a straight line, and the constant pace of \(2\) in \(x\) for every unit in \(t\) underlines the linear relationship.
Sketching these points provides a tangible understanding of the function's characteristics, helping students or anyone learning to connect numerical analysis with geometric representation.