Problem 17 - Algebra and Trigonometry

Question

Use point plotting to graph the plane curve described by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of \( t.\) \(x=t^{2}, y=t^{3} ;-\infty< t<\infty\)

Step-by-Step Solution

Answer

The graph of the plane curve described by the parametric equations \( x = t^2, y = t^3, -\infty < t < \infty \) happens to be a space-filling curve. Orientation is towards increasing \( x \) and \( y \) values as \( t \) increases.

Step 1: Understanding the Parametric Equations

The given parametric equations are \( x = t^2 \) and \( y = t^3 \). This means that for every value of \( t \), a corresponding point on the curve can be determined by squaring the value for \( x \), and cubing the value for \( y \).

Step 2: Table of Values

Set up a table of values to pinpoint representative points on the graph. Choose a series of values for \( t \) (which can be negative, zero, and positive). Find the corresponding \( x \) and \( y \) values using the given equations. For example, if \( t=-2 \), then \( x=4 \) and \( y=-8 \). If \( t=0 \), then \( x=0 \) and \( y=0 \). If \( t=2 \), then \( x=4 \) and \( y=8 \).

Step 3: Plotting the Points

Plot these points on a graph. Notice that as the value of \( t \) increases, both the \( x \) and \( y \) values increase as shown by the orientation of the curve.

Step 4: Drawing the Curve

Connect these points with smooth curves to have a complete picture of the graph. It's important to show the direction of the curve based on increasing \( t \) values. This can be represented by an arrow on the graph.

Key Concepts

Point Plotting

Plotting points is a crucial skill in graphing parametric equations. It involves placing points on the coordinate plane based on specified values of parameters. In this exercise, points are plotted by evaluating the parametric equations

• For each value of the parameter \( t \), compute corresponding \( x \) and \( y \) coordinates using the equations \( x = t^2 \) and \( y = t^3 \).
• Prepare a table of values where each row represents a different \( t \) value and the resulting \( x, y \) values arededuced.
• Plot these \( (x, y) \) pairs on the coordinate plane.

Values of \( t \) can be:- Negative: For example, if \( t = -2 \), the point is \((4, -8)\).- Zero: If \( t = 0 \), the point is \((0, 0)\), right at the origin.- Positive: For example, \( t = 2 \) results in the point \((4, 8)\).
By carefully choosing some values and extending to others, a pattern will emerge that helps visualize the curve.

Graphing Techniques

Once the points are plotted, the next step is to visually connect them to form a continuous curve. Use the following steps:

• Start by plotting all the points obtained from point plotting on graph paper or using graphing software.
• Identify if there's an apparent direction or pattern that emerges as you graph these points.
• Draw a smooth curve through the plotted points, noting that the curve might twist and turn depending on the parametric relationship.
• In our exercise, as \( t \) increases, the curve swoops from one quadrant to another, reflecting changes as \( x \) and \( y \) grow larger.

Every parametric curve will have a direction, which is indicated using arrows. Arrows guide the eye along the path illustrating how the curve is traced by increasing \( t \). This helps showcase the orientation and is a significant part of graphing techniques for parametric curves.

Coordinate Systems

The coordinate system is the foundation of any graphing process. Here, we use the Cartesian coordinate system, consisting of horizontal (\( x \)-axis) and vertical (\( y \)-axis) lines. Parametric equations use this system to illustrate curves by defining points in terms of a parameter \( t \).
Key aspects include:

• Quadrants: The plane is divided into four quadrants by these axes, where any point can reside based on its \( x \) and \( y \) values.
• Positive and Negative Values: The Cartesian system supports plotting positive and negative points, enabling full representation of the curve. For example, negative \( t \) values can produce negative \( y \) values.
• Axes: Always label your axes clearly. Starting from known points helps define how the curve spans across the coordinate system visually.

The Cartesian system is integral, allowing for accurate representation of parametric relationships, and helping visualize how such equations behave.

Algebraic Expressions

Algebraic expressions in parametric equations define the relationship between variables using an additional parameter \( t \). Here, we have the expressions \( x = t^2 \) and \( y = t^3 \).
To understand these:

• Squaring and Cubing: Reflects different growth rates. While \( x \) grows at \( t^2 \) rate, \( y \) grows much faster due to \( t^3 \).
• Variable Dependencies: Both \( x \) and \( y \) depend on \( t \), revealing how different values impact the curve's shape.
• Simplification: While computing, simplify expressions to smaller integers or fractions, making point plotting easier.

Being comfortable with these expressions allows you to predict how changes in \( t \) will influence the curve's direction and shape. It bridges algebra with graphical interpretation, crucial for graphing mastery.