Problem 40 - Precalculus

Question

Use a graphing utility to graph the curve represented by the parametric equations. $$\begin{aligned} &x=|t+2|\\\ &y=3-t \end{aligned}$$

Step-by-Step Solution

Answer

The graph for the parametric equations \(x = |t+2|\) and \(y = 3 - t\) will have 'V' shape for \(x\) and a declining line for \(y\). These intersect to form a curve, which will exactly follow a straight line while entering and exiting from the apex of 'V'.

Step 1: Understand the Equations

The given parametric equations are \(x = |t+2|\) and \(y = 3 - t\). Here, \(x\) represents the absolute value of the quantity \(t + 2\), and \(y\) is linearly decreasing with \(t\). The absolute value will give a 'V' shape graph while the second equation will give a straight line.

Step 2: Draw the Graph of Each Equation Separately

Since the algebraic equation is hard to interpret directly, we'll first draw the graphs of the equations separately. Choose a range for \(t\) and calculate \(y\) and \(x\) respectively for each \(t\). Draw these graphs to better understand their behavior.

Step 3: Draw the Parametric Graph

Now, for the same values of \(t\), pair the corresponding \(x\) and \(y\) values. Each pair \((x, y)\) represents a point in the 2D space for each instant \(t\). Plot these points on the same graph to see the desired parametric curve.

Key Concepts

Graphing Utilities

Graphing utilities are valuable tools used to bring mathematical equations to life. They help us visualize equations by plotting them on a graph. This makes it easier to understand the behavior and shape of different mathematical functions. These utilities often come as software programs or handheld calculators.

Using a graphing utility for parametric equations involves inputting the individual equations, one for each dimension of the graph. In our case, the equations are given for both \(x\) and \(y\), and they depend on a parameter \(t\). The graphing utility plots these onto a coordinate system to show the trajectory in two-dimensional space.

Advantages of using graphing utilities include:

• Quick visualization of complex parametric curves, like the one generated from \(x = |t+2|\) and \(y = 3-t\).
• Ability to explore the effect of varying parameters easily.
• Immediate feedback, which is especially useful for learning and experimentation.

By allowing you to manipulate and view equations interactively, graphing utilities expedite the learning process and enhance comprehension.

Absolute Value

Absolute value is a fundamental concept in mathematics which denotes the distance of a number from zero on the number line. It is always a non-negative value. For example, the absolute value of both \(-5\) and \(5\) is \(5\).

In graphical terms, when you see an absolute value function involved, expect a specific kind of "V" shape in its graph. This is crucial in our parametric equation \(x = |t+2|\), which prominently influences the shape of our curve.

Key characteristics of absolute values include:

• For any real number \(a\), : \(|a| = a \) if \(a\) >= 0, and \(|a| = -a\) if \(a\) < 0.
• The function \(x = |t+2|\) indicates that as \(t\) crosses \(-2\), there’ll be a change in the slope on the graph, creating the typical "V" shape.

Recognizing the impact of absolute values in equations helps predict and interpret the structure of their graphs, especially when dealing with parameterized functions.

Linear Functions

Linear functions are among the simplest functions in mathematics and are characterized by a constant rate of change. A linear function can be expressed in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

In our parametric context, \(y = 3 - t\) represents a linear function. It shows how \(y\) changes linearly as \(t\) changes.

Key features of linear functions are:

• The graph of a linear function is always a straight line.
• The slope \(m\) indicates the steepness and direction of the line - positive for upward sloping, negative for downward sloping.
• The \(c\) value indicates where the line crosses the y-axis.

In dealing with our equation \(y = 3 - t\), understanding that \(t\) causes a decrease in \(y\) by a rate of \(1\) per unit increase in \(t\) is crucial. This allows us to predict and visualize the direction and slope of the line in the graph, even before using a graphing utility.