Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Match the ordered pair from Column II with the pair of parametric equations in Column I on whose graph the point lies. In each case, consider the given value of \(t\). I $$x=t^{2}+3, y=t^{2}-2 ; \quad t=2$$ II A. \((5,25)\) B. \((7,2)\) C. \((12,0)\) D. \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)

Step-by-Step Solution

Answer

The ordered pair (7,2) corresponds to option B.

Step 1: Identify the parametric equations

The parametric equations given in Column I are: \( x = t^2 + 3 \) and \( y = t^2 - 2 \). We also have the specific value for \( t = 2 \).

Step 2: Substitute the value of \( t \) into the equation for \( x \)

Replace \( t \) with 2 in the equation for \( x \):\[ x = (2)^2 + 3 = 4 + 3 = 7.\]

Step 3: Substitute the value of \( t \) into the equation for \( y \)

Now, replace \( t \) with 2 in the equation for \( y \):\[y = (2)^2 - 2 = 4 - 2 = 2.\]

Step 4: Form the ordered pair from the calculations

The calculations provide the point \((x, y)\) as \((7, 2)\).

Step 5: Match the calculated point with options in Column II

Find the matching ordered pair in Column II for \((7, 2)\), which is option B: \((7, 2)\).

Key Concepts

Ordered pairs and their significance

An ordered pair is a fundamental concept in coordinate geometry. It is represented as \((x, y)\) in a plane, where "x" is the horizontal coordinate and "y" is the vertical coordinate. This pair describes the exact position of a point on a Cartesian plane, allowing us to precisely map where each point lies.

Ordered pairs are pivotal when graphing parametric equations, as these are written in such a way that for every value of the parameter \(t\), there's a unique ordered pair \((x, y)\). In our problem, after substituting the given value of \(t\) into the parametric equations, we obtain a specific ordered pair. This pair, \((7, 2)\), represents a unique point on the graph corresponding to the given parametric equations. Identifying and matching this pair to a point in Column II helps conclude where the particular parametric point lies.

Understanding graphing in parametric equations

Graphing parametric equations involves plotting points derived from the equations as functions of a parameter, usually \(t\). Unlike simple x-y equations, parametric equations define both coordinates \(x\) and \(y\) using an independent parameter. This allows for more complex and flexible graphing patterns such as loops, circles, and other shapes that are difficult to graph using standard x-y methods.

The key function of graphing these equations is to trace the path of points described by the ordered pairs. By plotting each point where \( t \) is substituted, such as in the example \((7, 2)\) when \( t = 2 \), one can visualize the motion or how the graph is composed as \(t\) changes. Graphing helps us to understand not just stationary points, but how these points relate to each other and to the path of the complete graph.

Exploring the substitution method

The substitution method is crucial for working with parametric equations. This method involves replacing a variable (in this case, \(t\)) with a specific number to simplify equations and determine corresponding values for \(x\) and \(y\).

In our example, the substitution of \(t = 2\) into the parametric equations \(x = t^2 + 3\) and \(y = t^2 - 2\) provides a clear way to find the exact coordinates. By calculating \( x = (2)^2 + 3 \), and \(y = (2)^2 - 2\), we replaced all instances of \(t\), simplifying the equations to give concrete numbers: \( x = 7 \) and \( y = 2 \).

This method transforms an abstraction of potential values into specific, usable data—essential when trying to identify actual points on a graph.