Problem 15 - A Graphical Approach to Precalculus with Limits

Question

For each plane curve, (a) graph the curve, and (b) find a rectangular equation for the curve. $$x=t+2, y=t^{2} ; \text { for } t \text { in }[-1,1]$$

Step-by-Step Solution

Answer

Graph the points for \( x = 1, 2, 3 \) and corresponding \( y = 1, 0, 1 \); the equation is \( y = (x-2)^2 \).

Step 1: Understanding the Parametric Equations

The given parametric equations are:\[ x = t + 2 \]\[ y = t^2 \]with \( t \) in the range \([-1, 1]\). These describe how \( x \) and \( y \) change with respect to the parameter \( t \). Our goal is to graph this and find a relationship between \( x \) and \( y \) that eliminates \( t \).

Step 2: Graphing the Curve

To graph the curve, calculate the points for various values of \( t \) within the given range. Calculate the following:- For \( t = -1 \), \( x = -1 + 2 = 1 \) and \( y = (-1)^2 = 1 \)- For \( t = 0 \), \( x = 0 + 2 = 2 \) and \( y = 0^2 = 0 \)- For \( t = 1 \), \( x = 1 + 2 = 3 \) and \( y = 1^2 = 1 \)Plot these points on the plane and connect them smoothly, noting that as \( t \) changes from -1 to 1, \( y \) will form a parabolic shape.

Step 3: Eliminating the Parameter t

To find a rectangular equation, express \( y \) in terms of \( x \). From the equation \( x = t + 2 \), solve for \( t \):\[ t = x - 2 \]Substitute \( t \) in the equation for \( y \):\[ y = (x - 2)^2 \]This gives us the rectangular equation of the curve.

Step 4: Verifying the Solution

Check the derived rectangular equation by substituting back the values of \( x \) and their corresponding \( y \) from the table in step 2. For point verification:- \( x = 1 \), \( y = (1-2)^2 = 1 \)- \( x = 2 \), \( y = (2-2)^2 = 0 \)- \( x = 3 \), \( y = (3-2)^2 = 1 \)These correspond with the original parametric equations confirming our solution is correct.

Key Concepts

Rectangular Equation

The concept of a rectangular equation is fundamental in converting a parametric equation involving an independent parameter into a single equation that describes the relationship between variables. In this exercise, we started with parametric equations:\[ x = t + 2 \] and \[ y = t^2 \], where the parameter \( t \) ranges from \(-1\) to \(1\).Our aim was to eliminate the parameter \( t \) and find a relationship solely in terms of \( x \) and \( y \). Earlier, we solved for \( t \) using the equation \( x = t + 2 \), resulting in \( t = x - 2 \). By substituting this into the equation for \( y \), we derived the rectangular equation: \[ y = (x - 2)^2 \].This new equation describes the curve in the Cartesian plane without referencing the parameter \( t \), hence 'rectangular' because it can be plotted directly on an XY-coordinate system.

Graphing Curves

Graphing a curve from parametric equations involves plotting points for various values of the parameter \( t \) and connecting these points smoothly.In the exercise, the parametric equations:\[ x = t + 2 \] and \[ y = t^2 \], guide the point calculations. With \( t \) ranging from \(-1\) to \(1\), we calculate key points:

• For \( t = -1 \), \( x = 1 \) and \( y = 1 \).
• For \( t = 0 \), \( x = 2 \) and \( y = 0 \).
• For \( t = 1 \), \( x = 3 \) and \( y = 1 \).

By plotting these points and visually connecting them, we see the characteristic shape of the parabola on the graph. Graphing this way helps visualize how the math describes the shape and trajectory of the curve.

Parameter Elimination

Eliminating a parameter transforms parametric equations into a straightforward Cartesian or rectangular form. This process simplifies graphing and interpreting the curve, as fewer steps are needed to see the relationship between the variables directly.To eliminate \( t \) from the parametric equations \( x = t + 2 \) and \( y = t^2 \), we first solve \( x = t + 2 \) for \( t \):\[ t = x - 2 \]. We then substitute \( t \) in the equation for \( y \), giving us the simplified form:\[ y = (x - 2)^2 \].This technique not only offers a clear view of how \( y \) relates to \( x \) but also makes graphing more direct and less reliant on calculating intermediate points with \( t \). This is particularly helpful in conveying the broader shape and behavior of curves.