Problem 5 - Precalculus

Question

Consider the parametric equations \(x=\sqrt{t}\) and \(y=3-t\) (a) Create a table of \(x\) - and \(y\) -values using \(t=0,1,2\) \(3,\) and 4 (b) Plot the points \((x, y)\) generated in part (a), and sketch a graph of the parametric equations. (c) Find the rectangular equation by eliminating the parameter. Sketch its graph. How do the graphs differ?

Step-by-Step Solution

Answer

For the given parametric equations, x values are \([0, 1, \sqrt{2}, \sqrt{3}, 2]\) and corresponding y values are \([3, 2, 1, 0, -1]\) for t values 0 through 4. The graph of the parametric equations is a downward-opening parabola, traced from left to right as t increases. The rectangular equation equivalent is \(y = 3 - x^2\), which also forms a downward-opening parabola. The key difference is the parametric representation includes directionality, tracing the progression of t, while the rectangular representation does not.

Step 1: Generate x and y values

Using given values of t (t=0, 1, 2, 3, and 4), calculate x and y values:For \(x = \sqrt{t}\) and \(y = 3 - t\):When \(t = 0\), \(x = 0, y = 3\)When \(t = 1\), \(x = 1, y = 2\)When \(t = 2\), \(x = \sqrt{2}, y = 1\)When \(t = 3\), \(x = \sqrt{3}, y = 0\)When \(t = 4\), \(x = 2, y = -1\)

Step 2: Plot the points

Take the \((x, y)\) pairs generated in the above step and plot them on a graph. Starting from (0,3) as t increases, the points descend down and to the right along a visually discernible curve.

Step 3: Transition to a rectangular equation

To move from the parametric equation to a rectangular one, eliminate the parameter t. \(x=\sqrt{t}\) can be rewritten as \(t=x^2\). Substitute \(x^2\) for \(t\) in the second equation to get \(y = 3 - x^2\).

Step 4: Sketch the rectangular graph

Using the rectangular equation \(y = 3 - x^2\), sketch a graph. This is a downward-opening parabola with a vertex at (0,3). As x values increase and decrease from 0, y values descend.

Step 5: Compare the graphs

Both the parametric graph (from Step 2) and the rectangular graph (from Step 4) share the same shape and orientation (a downward-opening parabola). However, the parametric graph includes directionality, showing the curve traced out as t increases. In contrast, the rectangular graph does not inherently convey this directional information.

Key Concepts

rectangular equation

In mathematics, a rectangular equation relates two variables, such as \(x\) and \(y\), using standard coordinate axes. Unlike parametric equations, rectangular equations do not involve a third variable or parameter. To find a rectangular equation from parametric equations, one typically eliminates the parameter (often \(t\)). Doing this emphasizes how \(x\) and \(y\) relate to each other directly. For example, from the given parametric equations \(x = \sqrt{t}\) and \(y = 3 - t\), we eliminate \(t\) by expressing \(t\) in terms of \(x\): \(t = x^2\). Substituting this into the \(y\) equation gives us the rectangular form \(y = 3 - x^2\). This equation clearly shows a relationship between \(x\) and \(y\) as a parabola, which is beneficial for graphing on standard coordinate axes without tracking a parameter.

graphing parametric equations

Graphing parametric equations involves plotting points based on functions that express one or both coordinates \((x, y)\) in terms of a third variable, the parameter \(t\). This added parameter creates an opportunity to visualize more complex shapes and curves than might initially appear with direct \(y\) vs. \(x\) plotting. In the example, the parametric equations \(x = \sqrt{t}\) and \(y = 3 - t\) allow for generating a series of \((x, y)\) points by inputting different \(t\) values.

• Start by choosing values for \(t\), which in the exercise were \(0, 1, 2, 3, \) and \(4\).
• Plug these into the parametric equations to get a matching \(x\) and \(y\).

By plotting these points, you can visualize the corresponding curve. This method is especially beneficial for visualizing how a curve is traced and understanding motion or transformation along the curve from increasing a plot table progressively with \(t\) values.

parameter elimination

Parameter elimination is a crucial technique used to convert parametric equations into a single rectangular equation. This strategy often simplifies the analysis and graphing of equations by reducing complexity, enabling one to analyze the relationship between \(x\) and \(y\) directly.In the given problem, we had \(x = \sqrt{t}\) and \(y = 3 - t\). To eliminate the parameter \(t\), we first express \(t\) in terms of \(x\) from \(x = \sqrt{t}\), resulting in \(t = x^2\). Next, substitute \(x^2\) back into the \(y\) equation: \[y = 3 - x^2\]This substitution removes the parameter, resulting in a rectangular equation entirely in terms of \(x\) and \(y\). The process of parameter elimination is not just a technique to convert equations, it simplifies them into a form that can be readily analyzed or visualized as a familiar graph, like a parabola.

plotting points

Plotting points is a fundamental process in graphing parametric or rectangular equations. For parametric equations like the ones in the problem, the process starts with generating pairs of \((x, y)\) coordinates. These are computed by substituting selected values for \(t\) into the equations.

• When \(t = 0\), their result is \(x = 0\) and \(y = 3\).
• As \(t\) increases, the coordinated sequence descends, providing the sets: \((1,2)\), \((\sqrt{2},1)\), \((\sqrt{3},0)\) and \((2,-1)\).

Once these points are obtained, plot them on a graph. This visual representation aids in observing the shape traced by the curve as \(t\) changes.In our case, the points illustrate a distinctive trajectory forming a parabola. This approach, particularly in parametric equations, adds an extra layer of understanding, discerning not only the curve's shape but also the directional flow defined by increases of \(t\). By plotting, you ensure an accurate and intuitive visualization of how the values interconnect to form the overall graph.