Problem 36 - Precalculus

Question

Use a graphing utility to graph the curve represented by the parametric equations. $$\begin{aligned} &x=t+4\\\ &y=t^{2} \end{aligned}$$

Step-by-Step Solution

Answer

The graph of the given parametric equations is a parabola opening upwards and shifted 4 units to the left.

Step 1: Identify the parametric equations

The parametric equations given are \(x = t + 4\) and \(y = t^2\). These equations represent the x and y coordinates of the points on the curve respectively, where t is the parameter.

Step 2: Insert the equations into the graphing utility

The next step is to input these equations into a graphing utility. For example, if you use a tool like Desmos or GeoGebra, input the equations as they are.

Step 3: Set the range of values for the parameter t

Set a range of values for t. For a standard curve, it's common to use a range from -10 to 10. The exact range might depend on the specific graphing utility used, but in general this range should give a good picture of the curve.

Step 4: Graph the curve

Once the equations and the range for t are set, graph the curve. It should display parabolic behavior, opening upwards, but shifted 4 units to the left due to the \(x = t + 4\) equation.

Key Concepts

Understanding Graphing Utilities

When visualizing parametric equations, a graphing utility is an invaluable tool. These utilities, such as Desmos or GeoGebra, allow us to input equations directly and see their graphical representations instantly. Graphing utilities have several advantages:

• They help in understanding complex mathematical concepts by providing visual aids.
• You can dynamically manipulate parameters and see how the graph changes in real-time.
• Results can be instantly generated, saving time on manual plotting.

To use these tools effectively, simply input the parametric equations you are working with. For instance, in this exercise, you would enter the equations directly into the utility, ensuring that they are correctly formatted for the software being used. Once entered, these programs will automatically plot the corresponding curve based on the given parameter range.

Determining the Range of Parameters

Determining a suitable range for parameters in parametric equations is crucial for accurately graphing the curve. The parameter, often denoted as \(t\), is a variable that both \(x\) and \(y\) depend on.

Selecting a range requires consideration of several factors:

• The general behavior of the functions involved. For example, if \(t\) falls within \(-10\) to \(+10\), it usually provides a comprehensive view.
• Understanding the desired section of the curve you wish to visualize. Some graphs may have interesting features outside the standard range, necessitating adjustments.
• The graphing utility being used might have its limits; it is vital to ensure the range is supported by the tool.

By carefully selecting the range of \(t\), you'll be able to graph all the essential parts of the curve, making sure you don't miss important behaviors or trends.

Plotting Coordinate Points from Parametric Equations

Parametric equations represent curves by defining \(x\) and \(y\) as functions of a third variable, typically \(t\). To plot the curve, one needs to generate coordinate points using different values of \(t\).

Here’s a simple process for plotting these points manually:

• Choose several values for \(t\) within the specified range.
• Substitute each value of \(t\) into the \(x = t + 4\) equation to get the \(x\)-coordinate.
• Do the same for the \(y = t^2\), providing the \(y\)-coordinate.

Once you have a collection of \( (x, y) \) points, plot these on a graph to see the curve.

Using a graphing utility automates this process, making it easier to visualize the curve by calculating and plotting points for many values of \(t\) simultaneously. This yields a smooth curve and helps in understanding the relationship between the parameters and their real-world implications.