Problem 62 - 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. \(x=t^{2}-t+6, y=3 t\)

Step-by-Step Solution

Answer

The sketch of the curve is an upward-opening parabola shifted by 6 units to the right. The domain is \([4, \infty)\) and the range is \((- \infty, \infty)\).

Step 1: Express x and y in terms of t

The given parametric equations are \(x = t^{2}-t+6\) and \(y = 3t\). These equations already express \(x\) and \(y\) in terms of \(t\).

Step 2: Plot points for the curve

Substitute various values of \(t\) into the above equations to get corresponding values of \(x\) and \(y\). For instance, you can choose values ranging from -2 to 2. For each value of \(t\), calculate \(x\) using \(x = t^{2}-t+6\) and \(y\) using \(y = 3t\), then plot these points on a 2D graph.

Step 3: Draw the planar curve

Join the plotted points to draw the planar curve. Note that as \(t\) increases or decreases, the graph will extend to the right or the left, respectively, and upwards or downwards. Thus, the graph does not have curves.

Step 4: Determine the domain and range

The domain is the set of all possible x-values, while the range is the set of all possible y-values. Notice from the graph that \(x\) can take any value greater than or equal to 4, while \(y\) can take any real number. Thus, the domain is \([4, \infty)\) and the range is \((- \infty, \infty)\).

Key Concepts

Sketching Plane Curves

The concept of sketching plane curves with parametric equations is like translating a language of numbers into a visual story. Imagine each point of a curve as a momentary snapshot of a moving point on the graph. To bring the curve to life, you would:

• Map out several positions of this moving point by choosing a range of 't' values. These can be positive, negative, or zero.
• Calculate corresponding 'x' and 'y' coordinates using the given parametric equations.
• Mark these coordinates on a two-dimensional graph as if you were plotting points for a treasure map.
• Connect the dots smoothly, considering how the curve behaves as 't' moves from negative to positive values. Does the curve turn? Does it make a loop, or does it travel off towards infinity?

In the given exercise, the set of equations creates a specific path defined by the values of 't'. Here, as 't' increases or decreases, the curve’s position adjusts accordingly, painting a unique shape in the plane that tells its own mathematical story.

Parametric Equations Domain and Range

The domain and range of a set of parametric equations are akin to the boundaries of a playground; they tell us how far the values of 'x' and 'y' can go given the parametric equations. For the domain, we look at all possible 'x' values, and for the range, the focus is on 'y' values.

• The domain is like thinking about the horizontal confines of our moving point. How far left or right can it go on the x-axis?
• The range is about the vertical limits. Imagine a ladder — how high and low can the point climb on the y-axis?

In this exercise, after plotting the points and visualizing the behavior of the curve, we conclude that the x values start at 4 and climb upwards without end as 't' gets larger—thus, the domain is \[4, \infty)\]. Meanwhile, 'y' travels up and down the entire y-axis unrestricted, so the range is \[−\infty, \infty\]. The use of interval notation expresses these concepts neatly, specifying the playground for both 'x' and 'y' values.

Plotting Parametric Equations

Plotting parametric equations is about turning abstract algebraic expressions into a concrete visual representation. Here’s how you make this translation happen:

• Pick a selection of 't' values to cover a range you’re interested in. You can think of 't' as time in a movie. What happens at each timestamp?
• Input these 't' values into your parametric equations to find the corresponding 'x' and 'y' coordinates, like reading a script to find out where the actors stand in each scene.
• Take these coordinate pairs and put them on the graph, like placing props on a stage.
• When you've got enough points, draw a curve through them, making a path as smooth as possible. This is like connecting scenes to make a full movie.

The exercise demonstrates this with the parametric equations \(x = t^{2}-t+6\) and \(y = 3t\). By selecting values such as -2, -1, 0, 1, and 2 for 't', calculating the corresponding 'x' and 'y', and then plotting them, a curve emerges. The smooth connection of these individual 'scenes' gives a complete picture of how the curve behaves across the range of 't' values.