Problem 5 - A Graphical Approach to College Algebra

Question

Graph each pair of parametric equations by hand, using values of t in \([-2,2] .\) 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. $$x=t^{2}+2, \quad y=-t+1$$

Step-by-Step Solution

Answer

Plot points (6,3), (3,2), (2,1), (3,0), (6,-1) and connect with a smooth curve.

Step 1: Set Up the Table

Create a table with three columns: one for the values of \(t\), one for \(x(t)\), and one for \(y(t)\). We will use the \(t\) values from \(-2\) to \(2\).

Step 2: Calculate x(t) Values

Use the equation \(x = t^2 + 2\) and compute \(x\) for each \(t\) value:- For \(t = -2\): \(x = (-2)^2 + 2 = 4 + 2 = 6\)- For \(t = -1\): \(x = (-1)^2 + 2 = 1 + 2 = 3\)- For \(t = 0\): \(x = 0^2 + 2 = 2\)- For \(t = 1\): \(x = 1^2 + 2 = 1 + 2 = 3\)- For \(t = 2\): \(x = 2^2 + 2 = 4 + 2 = 6\).

Step 3: Calculate y(t) Values

Use the equation \(y = -t + 1\) and compute \(y\) for each \(t\) value:- For \(t = -2\): \(y = -(-2) + 1 = 2 + 1 = 3\)- For \(t = -1\): \(y = -(-1) + 1 = 1 + 1 = 2\)- For \(t = 0\): \(y = -0 + 1 = 1\)- For \(t = 1\): \(y = -1 + 1 = 0\)- For \(t = 2\): \(y = -2 + 1 = -1\).

Step 4: Record the Coordinate Pairs

Write down the coordinate pairs \((x, y)\) for each corresponding \(t\) value:- \(t = -2\): \((6, 3)\)- \(t = -1\): \((3, 2)\)- \(t = 0\): \((2, 1)\)- \(t = 1\): \((3, 0)\)- \(t = 2\): \((6, -1)\).

Step 5: Plot the Points

Plot the coordinate pairs on a graph paper. Locate each pair on the Cartesian plane by tracing the \(x\) value horizontally and the \(y\) value vertically.

Step 6: Join the Points

Connect the plotted points with a smooth line or curve, following the order of the \(t\) values. Since \(x\) is symmetrical for \(t = -2\) and \(t = 2\) and also for \(t = -1\) and \(t = 1\), you'll notice symmetry in the shape of the graph.

Key Concepts

Coordinate System

The coordinate system is a fundamental tool in graphing equations. It's essentially a grid where you plot points, offering a visual way to understand mathematical relationships. Each point on this grid is defined by a pair of numbers, called coordinates:

• The first number is the x-coordinate, determining the horizontal position.
• The second is the y-coordinate, positioning the point vertically.

To graph parametric equations, like the given set

• \(x = t^2 + 2\) and
• \(y = -t + 1\),

you'll find x and y for different values of the parameter \(t\). This way, you don't directly relate x and y but use a third variable, t, which helps create a curve on this system. Understanding where these points lie in the plane is the first step in graphing them by hand.

Graphing by Hand

Graphing by hand is a skill that builds a deeper understanding of mathematical concepts. It requires converting parametric equations into points that you can plot. Here’s a simple way to tackle this:

• Start by selecting values for the parameter \(t\). In this case, \([-2, -1, 0, 1, 2]\) are great choices.
• Calculate corresponding \(x\) and \(y\) values by substituting these \(t\) values into your equations.
• Once you’ve done the calculations, write down each coordinate as a pair \((x, y)\).

For instance, with \(t = -2\), we find \(x = 6\) and \(y = 3\), giving us the point \((6, 3)\). Repeat this for all values of \(t\).

Finally, plot each of these points on a graph and join them with a smooth line or curve. Practicing graphing manually helps solidify core concepts in math and strengthens your problem-solving skills.

Symmetry in Graphs

Symmetry is a key concept in understanding graphs. It helps identify patterns and makes graphing simpler. With parametric equations, symmetry often answers questions about the nature of the graph without plotting every single point.In the given problem, we see that:

• The value of \(x\) was 6 for both \(t = -2\) and \(t = 2\).
• Similarly, \(x\) was 3 for both \(t = -1\) and \(t = 1\).

This is an example of reflection symmetry across the y-axis, showing that our graph will mirror itself on this axis. Recognizing this mirroring can save time and energy when graphing by hand. It acts as a shortcut, confirming your plot as you sketch it out. Looking for symmetry in graphs not only supports accuracy in drawing but also builds a solid understanding of how functions behave visually.