Problem 61 - Algebra and Trigonometry

Question

Sketch the plane curve represented by the given parametric equations. Then use interval notation to give each relation’s domain and range. \(x=t^{2}+t+1, y=2 t\)

Step-by-Step Solution

Answer

The parametric curve represented by the equations is sketched by the parabolic function \(x=y^2/4 + y/2 + 1\). The domain and range of these equations, using interval notation, are both \(-\infty, +\infty\).

Step 1: Sketch the parameterized curve

In order to properly sketch the curve, we will express 'y' as a function of 'x'. By manipulating the second equation \(y = 2t\), we get \(t = y/2\). We can substitute this expression for 't' into the first parametric equation to get the function \(x=y^2/4 + y/2 + 1\). Plotting this function will provide the graph of the curve.

Step 2: Domain and Range

The domain of a function is the complete set of possible values of the independent variable (in this case 't'). Here, 't' has not been limited, so it extends all the way from negative infinity to positive infinity; its domain will thus be \(-\infty, +\infty\). As for the range, looking at the equation \(y=2t\), as 't' approaches +\(\infty\), y will also trend towards +\(\infty\). Similarly, as 't' approaches -\(\infty\), y will also trend towards -\(\infty\). Therefore, the range of this function is also \(-\infty, +\infty\)

Key Concepts

Domain and Range

When working with parametric equations, it's crucial to identify the domain and range of each function, which are the set of possible inputs and outputs, respectively. In parametric equations like the ones given, where \( x = t^2 + t + 1 \) and \( y = 2t \), the independent variable 't' is not restricted, meaning it can take any real number value. Consequently, the domain of the parameter 't' is from \(-\infty\) to \(+\infty\).

For the range, it's important to observe the values that the functions for 'x' and 'y' can output. Since 'y' is a straightforward function of 't' (\( y = 2t \)), and 't' encompasses all real numbers, 'y' will also span from \(-\infty\) to \(+\infty\). As for 'x', when you manipulate its equation in terms of 'y', \( x = \frac{y^2}{4} + \frac{y}{2} + 1 \), you see 'x’ is determined by values of 'y', which again range from \(-\infty\) to \(+\infty\). Hence, the range of 'x' is also all real numbers.

Plane Curves

In mathematics, a plane curve is a curve that lies on a flat, two-dimensional plane. Parametric equations are a powerful tool in describing these plane curves because they allow you to express both x and y coordinates as functions of a third variable, often 't'.

In the given equations, \( x = t^2 + t + 1 \) and \( y = 2t \), each value of 't' provides a specific point on the curve. This forms a continuous path as 't' changes. Plane curves can represent a vast variety of shapes, from circles to parabolas, depending on the actual functions involved. They can be smooth, continuous, or even have sharp turns or loops.

Parametric curves are often used in physics and engineering to describe motions where time is a key factor, making them an essential concept in understanding dynamic systems. They help visualize how two quantities change simultaneously with respect to a third parameter.

Sketching Graphs

Sketching graphs of parametric equations involves plotting points given by expressions for 'x' and 'y' as functions of the parameter 't'. Here's a simple way to approach it for any parametric curve:

• First, find a few key points by substituting simple values of 't' into the equations for 'x' and 'y'. This gives some starting points that indicate the shape of the curve.
• Next, derive y as a function of x if possible, like we did with \( x = \frac{y^2}{4} + \frac{y}{2} + 1 \), to understand how the curve behaves overall.

When sketching the provided equations, begin by noting significant values, such as where \( t = 0 \), \( t = 1 \), and \( t = -1 \) to identify its basic structure. This helps illustrate how the curve is positioned and oriented on the plane.

Then, consider the behavior at extreme values of 't' (as it goes towards positive or negative infinity). This ensures you capture the full extent of the graph. Finally, connect the points smoothly to form a cohesive, accurate representation of the curve. This visualization is not just an academic exercise; it’s a useful step in applications, from computer graphics to analyzing real-world motion.