Problem 39 - Precalculus

Question

Use a graphing utility to graph the curve represented by the parametric equations. $$\begin{array}{l} x=2 t \\ y=|t+1| \end{array}$$

Step-by-Step Solution

Answer

The graph is a V shape curve, with the point of the V at x=-2. For x< -2, the graph slopes downwards while for x > -2, the graph slopes upwards.

Step 1: Analyze the parametric equations

We are given two equations. First equation determines the x coordinates and it's given by \(x = 2t\), meaning x varies linearly with parameter t by a factor of 2. The second one, \(y = |t+1|\), determines the y coordinates. This is an absolute value equation, which means y will always be positive or zero. It makes the graph have different forms depending on whether the expression inside the absolute value sign is positive or negative.

Step 2: Convert the absolute value equation into a piecewise function

We convert \(y = |t+1|\) into a piecewise function. It results in two separate functions: \(y = t+1\) for \(t \geq -1\) and \(y = -t -1\) for \(t<-1\). This means for values of t greater than -1, we have an upwards sloping line, otherwise, we have a downward sloping line.

Step 3: Plot the parametric function

Using a graphing utility, we set the x-values to be \(x = 2t\), and plot the above piecewise function for y coordinate. The graph should result in a V shaped curve. The change occurs at the value of \(t = -1\), which corresponds to \(x = -2\), since x is twice the t value.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that helps visualize complex mathematical equations and functions. These can range from handheld calculators to sophisticated software available on computers. In our problem, the graphing utility assists in visualizing the parametric equations:

• x = 2t
• y = |t+1|

Understanding how to use a graphing utility effectively is crucial for exploring mathematical concepts beyond their algebraic expressions. By inputting the parametric equations, the graphing utility provides a visual representation of the curve, making it easier to comprehend its behavior and features.
When using a graphing utility, always ensure you set an appropriate range for the parameter (t in this case) to capture the entire curve of interest. This allows you to observe where the function changes its behavior, which is particularly observable with functions like the absolute value, where the graph takes on a V-shaped form.

Piecewise Functions

Piecewise functions are an excellent way to represent expressions that have different behaviors over various intervals. The parametric equation for y in our exercise is initially given as an absolute value function:

• y = |t+1|

To convert it into a piecewise function, we break it into parts. This is done by considering the expression inside the absolute value:

• If \(t \geq -1\), then the function behaves as \(y = t+1\).
• If \(t < -1\), the function behaves as \(y = -t-1\).

Breaking it down in this way makes it much easier to graph or analyze by hand. You can see these two parts in your graph as two lines meeting at a point, which is where \(t = -1\). Understanding piecewise functions is crucial as it allows you to handle more complicated functions by examining them section by section.

Absolute Value Functions

Absolute value functions are fundamental in mathematics and they often crop up in real-world problems. Here, we look at the absolute value equation of the form:

• y = |t+1|

The absolute value operator makes any negative input positive, which affects how the graph looks. In this case, for any value of \(t\), \(|t+1|\) will produce a non-negative result, leading to the graph having a point of symmetry over the line \(t = -1\).
Understanding absolute value functions involves recognizing that they create a V-shaped graph. This occurs because the absolute value function reflects negative outputs into positive ones around a specified point, known as the vertex. Breaking an absolute value function into a piecewise form, as in our solution, allows you to see the two linear components making up the V shape and understand how the function changes behavior at the vertex.