Problem 4 - 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. $$x=t-1, \quad y=t^{2}+2$$

Step-by-Step Solution

Answer

Plot points (-3, 6), (-2, 3), (-1, 2), (0, 3), (1, 6) and join with a parabola.

Step 1: Set Up Table for t, x, and y Values

Begin by setting up a table with three columns labeled: \(t\), \(x\), and \(y\). We will evaluate both \(x\) and \(y\) for \(t = -2, -1, 0, 1,\) and \(2\).

Step 2: Compute x Values

Using the equation \(x = t - 1\), substitute each \(t\) value:- For \(t = -2\), \(x = -2 - 1 = -3\).- For \(t = -1\), \(x = -1 - 1 = -2\).- For \(t = 0\), \(x = 0 - 1 = -1\).- For \(t = 1\), \(x = 1 - 1 = 0\).- For \(t = 2\), \(x = 2 - 1 = 1\).Record these values in the \(x\) column of the table.

Step 3: Compute y Values

Using the equation \(y = t^2 + 2\), substitute each \(t\) value:- For \(t = -2\), \(y = (-2)^2 + 2 = 4 + 2 = 6\).- For \(t = -1\), \(y = (-1)^2 + 2 = 1 + 2 = 3\).- For \(t = 0\), \(y = 0^2 + 2 = 0 + 2 = 2\).- For \(t = 1\), \(y = 1^2 + 2 = 1 + 2 = 3\).- For \(t = 2\), \(y = 2^2 + 2 = 4 + 2 = 6\).Record these values in the \(y\) column of the table.

Step 4: Complete x, y Table

Summarize the table with columns for \(t\), \(x\), and \(y\):| \(t\) | \(x\) | \(y\) || ---- | ---- | ---- || -2 | -3 | 6 || -1 | -2 | 3 || 0 | -1 | 2 || 1 | 0 | 3 || 2 | 1 | 6 |

Step 5: Plot the Points

Plot the points \((-3, 6), (-2, 3), (-1, 2), (0, 3), (1, 6)\) on a coordinate plane based on the \(x\) and \(y\) values obtained.

Step 6: Join the Points

After plotting the points, draw a smooth curve through them to represent the graph of the parametric equations over the interval \([-2, 2]\). The points should form a parabola opening upwards.

Key Concepts

Graphing Parametric Equations

When tackling parametric equations, it's important to understand that these equations express coordinates in terms of a third variable, typically "t" for time. In this exercise, we use parametric equations to describe both the x and y coordinates based on t. For example, the equations given are:

• \(x = t - 1\)
• \(y = t^2 + 2\)

These equations indicate that as t changes, the location (x, y) on a coordinate plane changes, allowing us to trace a path or trajectory. Graphing parametric equations helps visualize the relationship between variables in a dynamic way. Begin by setting up a table that contains your chosen t-values and compute the matching x and y values for each t. This will generate specific points representing segments of a curve on the graph.

Coordinate Plane

The coordinate plane is a universal framework used in algebra and calculus to plot points, lines, and curves. In this exercise, the coordinate plane serves as our canvas to clearly see the parametric relationship between the variables. Fundamentally, it consists of two number lines: one horizontal (x-axis) and one vertical (y-axis), intersecting at a point called the origin.

• X-axis: Represents the horizontal component, in this case defined by \(x = t - 1\)
• Y-axis: Represents the vertical component, in this case defined by \(y = t^2 + 2\)

Plotting points obtained from evaluating these equations at specified t-values helps create a visual model. By doing this, we can appreciate not just where data points lay but also the form and trajectory of the path described by the parametric equations.

Curve Plotting

Curve plotting is essential for visualizing parametric equations on the coordinate plane. Once you've plotted the calculated points based on the parametric equations, the next step is to connect the dots. This action will reveal the entire curve represented by the equations, often resulting in a physiological shape like a parabola or ellipse.In this exercise, you begin by plotting the points

• \((-3, 6), (-2, 3), (-1, 2), (0, 3), (1, 6)\)

Next, join these points with a smooth curve. For this particular exercise, the curve forms an upward-opening parabola. This shape arises from the consistent quadratic relationship in the y-equation. Such visual representation aids in understanding how changes in t affect movement across the plane.

Parametric Form

Parametric form is a method used to express a set of equations where each variable in the system is expressed in terms of an independent parameter. It is particularly useful in fields like physics and engineering for describing trajectories and other paths.In this context, the parametric form involves equations \(x = t - 1\) and \(y = t^2 + 2\). Here, t acts like a clock, guiding the transformation of the values into the coordinate system. By systematically adjusting "t," you can observe how x and y evolve in response. This form grants a flexible, sequential capture of motion or changes linked dynamically rather than just statically as you have with regular Cartesian coordinates. It serves as a more holistic way to visualize complex relationships in mathematical problems.