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

Step-by-Step Solution

Answer

Plot points: \((-3, -4)\), \((-1, -3)\), \((1, -2)\), \((3, -1)\), \((5, 0)\) and connect with a line.

Step 1: Create a Table of Values

List the values of \( t \) you will use: \( t = -2, -1, 0, 1, 2 \). For each \( t \), calculate \( x \) and \( y \) using the given parametric equations.\( x \) is computed as \( x = 2t + 1 \) and \( y \) is computed as \( y = t - 2 \). Populate the table with \( (t, x, y) \) values obtained from substituting each \( t \) value in the equations.

Step 2: Fill in the Table

Substitute each \( t \) into the equations and fill in the table:- For \( t = -2 \): \( x = 2(-2) + 1 = -4 + 1 = -3 \) and \( y = -2 - 2 = -4 \) - For \( t = -1 \): \( x = 2(-1) + 1 = -2 + 1 = -1 \) and \( y = -1 - 2 = -3 \) - For \( t = 0 \): \( x = 2(0) + 1 = 0 + 1 = 1 \) and \( y = 0 - 2 = -2 \) - For \( t = 1 \): \( x = 2(1) + 1 = 2 + 1 = 3 \) and \( y = 1 - 2 = -1 \) - For \( t = 2 \): \( x = 2(2) + 1 = 4 + 1 = 5 \) and \( y = 2 - 2 = 0 \). Completed Table:\[\begin{array}{|c|c|c|}\hline t & x & y \\hline -2 & -3 & -4 \ -1 & -1 & -3 \ 0 & 1 & -2 \ 1 & 3 & -1 \ 2 & 5 & 0 \\hline\end{array}\]

Step 3: Plot the Points

Using the table of values, plot each point \((x, y)\) on a coordinate plane. The points to plot are: \((-3, -4)\), \((-1, -3)\), \((1, -2)\), \((3, -1)\), and \((5, 0)\).

Step 4: Connect the Points

Draw a smooth curve or straight line through the points plotted. Since these points are derived from linear parametric equations, they will form a straight line. Verify that the connections appear linear and pass through each plotted point.

Key Concepts

Plotting Points

Plotting points is a simple yet essential part of mathematics, especially when dealing with parametric equations. It involves marking specific coordinates on a graph to visualize an equation's solution. Let's break it down so it’s easy to follow:

To plot points, you need the coordinates, typically in - 2D space (x, y)
Each point is a location on the Cartesian coordinate system.
- The horizontal line is the x-axis. - The vertical line is the y-axis.
When plotting, start by finding your x-coordinate on the x-axis, and then move up or down to your y-coordinate on the y-axis.
In our exercise, we plotted the points - (-3, -4) - (-1, -3) - (1, -2) - (3, -1) - (5, 0)
These points were derived from the given parametric equations, and marking them correctly on the graph helps give a visual representation of those equations.

Table of Values

A table of values is a helpful tool for organizing data. It makes it easier to see patterns or to plot points on a graph. Here’s how to make and use a table of values efficiently:

1. **Choose values for the independent variable**: For parametric equations, this is often parameter t.
2. **Calculate corresponding dependent variables**: In our case, x and y, which are functions of t.
3. **Organize the data** into a table structure. For example:

• t = -2: x = -3, y = -4
• t = -1: x = -1, y = -3
• t = 0: x = 1, y = -2
• t = 1: x = 3, y = -1
• t = 2: x = 5, y = 0

This method allows you to efficiently find and list all needed coordinates for further analysis or graph plotting. After populating this table, proceed to plot each (x, y) on the graph.

Linear Equations

Linear equations are equations of the first degree, where the variables are not raised to any power other than one. They form straight lines when plotted on a graph, making them quite predictable and easy to work with.

Here’s what makes an equation 'linear':

• It can usually be written in the format y = mx + b, where m is the slope and b is the y-intercept.
• For parametric equations, each parameter t will give a pair of x and y values which align along a straight trajectory.
• In our exercise, the x and y generated from the parametric equations also result in a linear relationship - this means the plotted points will lie on a straight line.

Moreover, knowing that these underlying parametric equations represent linear functions helps us anticipate that the plotted points will naturally connect in a straight line, thereby allowing us to draw a single line through them confidently.