Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) sketch the curve represented by the parametric equations (indicate the orientation of the curve). Use a graphing utility to confirm your result. (b) Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. $$x=|t-1|, y=t+2$$

Step-by-Step Solution

Answer

The rectangular equation that represents the curve of the given parametric equations is \(x = |y - 3|\), which is valid for all values of \(y\). The graph is in a 'V' shape that opens horizontally with its vertex at (3,0).

Step 1: Sketch the Curve

To sketch the curve, evaluate the parametric equations for several values of \(t\). This will help in knowing what points to plot on the coordinate plane. Let's start by evaluating the pair of equations at \(t=-1, 0, 1, 2\). On substituting in the equations, we get 4 points: (2,1), (1,2), (0,3), (1,4). Now plot these points on a graph. Based on these points, the graph is in the first and second quadrants. As \(t\) increases, the curve moves upward and 'bounces' off the \(y\)-axis at some point. After sketching, use a graphing utility to verify.

Step 2: Eliminate the Parameter and Write Rectangular Equation

To eliminate the parameter, take note of the defined range of \(t\): \(y = t + 2 \rightarrow t = y - 2\). Now substitute \(t\) in the equation for \(x\) to get \(x = |y - 2 - 1|\) simplifying to \(x = |y - 3|\). This is the rectangular equation that represents the curve. The domain of this equation needs to be adjusted as original parametric equations don't have any range restriction. So the rectangular equation is valid for all \(y\).

Step 3: Summary

The graph represents an absolute function in the form \(x = |y - k|\) where \(k = 3\), which gives a 'V' shape that opens horizontally. The vertex of the 'V' is shifted to (3,0) on the y-axis. Therefore, the graph represents all points \((x, y)\) satisfying the equation \(x = |y - 3|\).

Key Concepts

Rectangular Equation

The transition from parametric equations to a rectangular equation is a valuable algebraic maneuver in math, particularly when analyzing curves. In the given exercise, we had parametric equations where the variable t plays the role of a parameter which defines both x and y in terms of t. The process to eliminate the parameter involves solving one of the equations for t and substituting it into the other. This produces an equation in terms of x and y only, which is known as a rectangular equation because it can be graphed in a standard Cartesian coordinate plane. For the set of equations x = |t - 1|, y = t + 2, we found the rectangular equation to be x = |y - 3|. This technique simplifies the visualization of the curve and is particularly useful when a graphing utility is not at hand.

Sketching Curves

Sketching curves from parametric equations is a nuanced art and requires understanding the behaviour of the parameters involved. By selecting values for the parameter t, we can calculate the corresponding x and y to plot specific points on the curve. This eventual sketch gives insight into the general shape and orientation of the curve. The key steps involve calculating and plotting a variety of points to observe patterns and then connecting these points in a smooth manner. The exercise demonstrated this with the parametric equations x = |t - 1|, y = t + 2, where the sketch revealed a V-shaped curve reflecting the absolute value function in a horizontal orientation.

Absolute Value Function

The absolute value function is depicted graphically as a V-shaped curve. In the context of the exercise, the absolute value affected the x-coordinate and was written as x = |t - 1|. When translated into a rectangular equation, it became x = |y - 3|. The curve of an absolute value function typically has its vertex at the origin; however, transformations can shift it across the coordinate plane. In this instance, the vertex of the V is located at the point where y is equal to 3. In general, absolute value functions express the distance from a number to zero on a number line, which is why they are always non-negative. They play a significant role in various mathematical contexts, including engineering and economics, where non-negative constraints are essential.

Graphing Utility

A graphing utility is a powerful tool for visualizing complicated algebraic equations. It can confirm the sketches of curves that we draw by hand, providing an accurate depiction for verification. In this exercise, after sketching by hand, the graphing utility could be used to confirm the V-shape curve obtained from the parametric equations. Graphing utilities accept both parametric and rectangular forms of equations, which aids in understanding the relationship between the two. They are indispensable in educational settings when it comes to illuminating the curves represented by more intricate functions.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is the fundamental framework used to graph equations and visualize how they represent geometric shapes and curves. Composed of a horizontal axis (the x-axis) and a vertical axis (the y-axis), the coordinate plane allows for plotting points, lines, and curves in two dimensions. The importance of converting parametric equations into a rectangular form becomes evident here; once this is done, the equation's graph can be easily understood and analyzed in the familiar context of the Cartesian plane. The exercise involving x = |t - 1| and y = t + 2 converges on the concept that every point on the plane is an intersection of a unique pair of x and y values which can correspond to different values of the parameter t.