Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Graph each pair of parametric equations by hand, using values of tin \([-2,2] .\) Make a table of \(t_{*}, x_{*}\) and \(y\) -values, using \(t=-2,-1,0,1,\) and \(2 .\) Then plot the points and join them with a line or smooth curve for all values of \(t\) in \([-2,2] .\) Do not use a calculator. $$x=t+1, \quad y=t^{2}-1$$

Step-by-Step Solution

Answer

Plot the points and connect them to form a parabola.

Step 1: Set Up the Table

Create a table with three columns labeled \( t \), \( x \), and \( y \), and five rows corresponding to the values of \( t \) we will use: \(-2, -1, 0, 1,\) and \(2\).

Step 2: Calculate x-values

Using the equation \( x = t + 1 \), calculate \( x \) for each \( t \) value in the range:- For \( t = -2 \), \( x = -2 + 1 = -1 \)- For \( t = -1 \), \( x = -1 + 1 = 0 \)- For \( t = 0 \), \( x = 0 + 1 = 1 \)- For \( t = 1 \), \( x = 1 + 1 = 2 \)- For \( t = 2 \), \( x = 2 + 1 = 3 \)

Step 3: Calculate y-values

Using the equation \( y = t^2 - 1 \), calculate \( y \) for each \( t \) value:- For \( t = -2 \), \( y = (-2)^2 - 1 = 4 - 1 = 3 \)- For \( t = -1 \), \( y = (-1)^2 - 1 = 1 - 1 = 0 \)- For \( t = 0 \), \( y = (0)^2 - 1 = 0 - 1 = -1 \)- For \( t = 1 \), \( y = (1)^2 - 1 = 1 - 1 = 0 \)- For \( t = 2 \), \( y = (2)^2 - 1 = 4 - 1 = 3 \)

Step 4: Fill in the Table

Combine your results to complete the table:\[\begin{array}{|c|c|c|}\hline t & x & y \\hline-2 & -1 & 3 \-1 & 0 & 0 \ 0 & 1 & -1 \ 1 & 2 & 0 \ 2 & 3 & 3 \\hline\end{array}\]

Step 5: Plot the Points

Using the table, plot the points \((-1, 3)\), \((0, 0)\), \((1, -1)\), \((2, 0)\), and \((3, 3)\) on a graph. Each point corresponds to a \((x, y)\) pair derived from each \(t\) value.

Step 6: Draw the Curve

Once all points are plotted, smoothly connect the points with either a straight line segment or a curve as indicated by the plotted points' pattern. This forms a part of a parabola.

Key Concepts

Graphing Techniques

Graphing techniques refer to the methods used to visually represent equations or data. When dealing with parametric equations, you often graph these by identifying key points on the graph based on a parameter, usually denoted as \( t \). To start:

• Create a table that involves the parameter \( t \) and corresponding \( x \) and \( y \) values.
• Choose a suitable range for \( t \). In our example, we use \([-2, 2]\).
• Compute the \( x \) and \( y \) values using the given parametric equations. Here, it's \( x = t + 1 \) and \( y = t^2 - 1 \).
• Plot the calculated points \((x, y)\) on a coordinate graph.
• Finally, connect these points smoothly to reflect the relationship the equations depict.

This method helps visualize how \( x \) and \( y \) change with respect to \( t \), noting any curves or lines that form.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses algebraic equations to represent geometric shapes like lines, curves, and points in a 2D plane through coordinates like \((x, y)\). Here’s how you apply it to our problem:

• Parametric equations are translated into coordinate points by considering each parameter \( t \) as an instance.
• The calculated \( (x, y) \) pairs from the parametric equations are plotted in the 2D coordinate system.
• Each point on the plane represents a specific \( t \) value, allowing the graph of the parametric equations to unfold.

By plotting with parametric equations, you see not only the shape (like a line or curve) but also the path taken by the moving point in the plane, enhancing the understanding of the geometrical properties of the equations.

Parabolic Curves

A parabolic curve is a symmetrical, U-shaped curve. Parabolas are fundamental shapes seen in quadratic functions and parametric forms can often illustrate them. For our example:

• The equation \( y = t^2 - 1 \) itself describes a typical parabolic curve.
• The parameter \( t \) guides the curve's direction and shape on the graph.
• Our computed points, like \((1, -1)\) or \((3, 3)\), when plotted, and connected will form this familiar shape.

Understanding how the equations drive the parabola helps in visualizing how changes in \( t \) influence the curve—a key aspect in many real-world applications ranging from physics to economics. Parabolic curves originating from parametric equations also serve as a bridge in understanding the shift from algebraic equations to their geometric representations.