Problem 2 - 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. $$x=-t+1, \quad y=3 t+2$$

Step-by-Step Solution

Answer

Plot the points: (3,-4), (2,-1), (1,2), (0,5), (-1,8) and connect them with a curve.

Step 1: Understanding Parametric Equations

We are given the parametric equations: \(x = -t + 1\) and \(y = 3t + 2\). A parametric equation defines the values of \(x\) and \(y\) in terms of a third variable, \(t\). Our task is to compute the \(x\) and \(y\) values for selected values of \(t\) within the range \([-2,2]\).

Step 2: Create the Values Table

Create a table with three columns: \(t\), \(x\), and \(y\). Choose \(t = -2, -1, 0, 1, 2\). For each \(t\), calculate \(x\) and \(y\) using the parametric equations.

Step 3: Calculate for \(t = -2\)

Substitute \(t = -2\) into both parametric equations: \(x = -(-2) + 1 = 3\) and \(y = 3(-2) + 2 = -4\). Add the point \((-2, 3, -4)\) to the table.

Step 4: Calculate for \(t = -1\)

Substitute \(t = -1\) into the equations: \(x = -(-1) + 1 = 2\) and \(y = 3(-1) + 2 = -1\). Add the point \((-1, 2, -1)\) to the table.

Step 5: Calculate for \(t = 0\)

Substitute \(t = 0\) into the equations: \(x = -(0) + 1 = 1\) and \(y = 3(0) + 2 = 2\). Add the point \((0, 1, 2)\) to the table.

Step 6: Calculate for \(t = 1\)

Substitute \(t = 1\) into the equations: \(x = -(1) + 1 = 0\) and \(y = 3(1) + 2 = 5\). Add the point \((1, 0, 5)\) to the table.

Step 7: Calculate for \(t = 2\)

Substitute \(t = 2\) into the equations: \(x = -(2) + 1 = -1\) and \(y = 3(2) + 2 = 8\). Add the point \((2, -1, 8)\) to the table.

Step 8: Plot the Points on the Graph

Using the points calculated: \((3, -4)\), \((2, -1)\), \((1, 2)\), \((0, 5)\), and \((-1, 8)\), plot these coordinates onto a graph with \(x\) and \(y\) axes.

Step 9: Connect the Points

Connect the plotted points with a line or smooth curve to illustrate the motion described by the parametric equations as \(t\) increases from \(-2\) to \(2\).

Key Concepts

Graphing Techniques

Graphing parametric equations requires understanding the trajectory we wish to represent. Since parametric equations like \(x = -t + 1\) and \(y = 3t + 2\) define both \(x\) and \(y\) in terms of a third variable \(t\), our graphing task is about revealing how \(x\) and \(y\) change with \(t\). When plotting these, it helps to:

• Create a table that pairs specific \(t\) values with corresponding \(x\) and \(y\) values.
• Plot each \((x, y)\) point as an individual solution for the given \(t\).
• Use a smooth curve to connect the dots, showing how the graph flows with changing \(t\).

Graphing in this way reflects how the object or line moves in the Cartesian plane, illustrating sequences of positions over a specified interval.

Coordinate Transformation

Coordinate transformation is key when dealing with parametric equations, as we're transferring coordinates from a parametric space (defined by \(t\)) to the Cartesian plane (defined by \(x, y\)). For equations like \(x = -t + 1\) and \(y = 3t + 2\):

• Identify a range for \(t\), such as \([-2, 2]\), which acts as our input values.
• Transform each \(t\) by substituting it into the parametric equations to get the Cartesian coordinates \( (x, y) \).
• For instance, with \(t = -2\), compute \(x = 3\) and \(y = -4\), transforming parametric points into Cartesian points.

This process translates the movement from parametric constraints into a visual form easily interpretable on graphing axes.

Algebraic Computation

Algebraic computation in the context of parametric equations involves managing two primary equations to compute the desired coordinates. Each equation offers insights into different dimensions:

• The equation \(x = -t + 1\) adjusts values inversely to \(t\), meaning as \(t\) increases, \(x\) decreases.
• The equation \(y = 3t + 2\) scales \(t\) significantly, suggesting a faster progression or growth in the \(y\) direction compared to \(x\).
• Use substitution to evaluate each equation at specific \(t\) points, thus building a series of \(x, y\) pairs.

Understanding how each equation modifies \(t\) into \(x\) or \(y\) offers a clear, algebraic picture on how graphs translate abstract numbers into visual data.

Plotting Points

Plotting points derived from parametric equations on a graph involves a few simple, yet critical steps:

• Use the table of values you've calculated (like \((3, -4), (2, -1), \) and so on) to determine exact graph positions.
• Place each point precisely on a Cartesian plane. Labeling can be helpful to notate which \(t\), \((x, y)\) coordinates correlate.
• Connect these plotted points with a smooth curve, representing the trajectory of movement as \(t\) progresses from \(-2\) to \(2\).

This curve not only illustrates the direction of travel along the path but indicates the speed dictated by how close or far points are. It helps to give life to static equations by visualizing motion and progression.