Problem 3 - College Trigonometry

Question

In Exercises \(1-20\), plot the set of parametric equations by hand. Be sure to indicate the orientation imparted on the curve by the parametrization. $$ \left\\{\begin{array}{l} x=2 t \\ y=t^{2} \end{array} \text { for }-1 \leq t \leq 2\right. $$

Step-by-Step Solution

Answer

Plot points from (-2, 1) to (4, 4) and draw a parabolic path with arrows from left to right.

Step 1: Understand the Parametric Equations

The parametric equations given are \(x = 2t\) and \(y = t^2\). These equations describe a curve parametrically, with the parameter \(t\) taking values in the range of \(-1 \leq t \leq 2\). This means that as \(t\) changes in this interval, \(x\) and \(y\) will change accordingly, plotting out a path.

Step 2: Identify Key Points

Substitute values of \(t\) in the equations to get corresponding \(x\) and \(y\) coordinates. For example: - When \(t = -1\), \(x = 2(-1) = -2\) and \(y = (-1)^2 = 1\), giving the point (-2, 1).- When \(t = 0\), \(x = 2(0) = 0\) and \(y = 0^2 = 0\), giving the point (0, 0).- When \(t = 1\), \(x = 2(1) = 2\) and \(y = 1^2 = 1\), giving the point (2, 1).- When \(t = 2\), \(x = 2(2) = 4\) and \(y = 2^2 = 4\), giving the point (4, 4).

Step 3: Plot the Points

Plot these points on the coordinate plane: (-2, 1), (0, 0), (2, 1), and (4, 4). Ensure the axes are appropriately scaled to include all these points.

Step 4: Connect the Points to Form the Curve

Draw a smooth curve through the points to illustrate the path. Since \(x = 2t\) and \(y = t^2\), the curve is a parabola and is commonly known as a 'parabolic path' of \(t\).

Step 5: Indicate Direction of the Path

The parameter \(t\) increases from -1 to 2, so the orientation of the curve follows this direction. As \(t\) increases, \(x\) increases from -2 to 4, and \(y\) forms a parabola. Mark the direction with arrows along the curve, moving from the left to the right.

Key Concepts

Parametrization

One of the key concepts in the study of plane curves is parametrization. It involves expressing points on a curve in terms of a parameter, typically denoted as \( t \). Essentially, this means using equations where \( x \) and \( y \) are written separately in terms of \( t \). This allows for a unified description of complex shapes and paths. For example, in the exercise above, the parametrization \( x = 2t \) and \( y = t^2 \) defines each point on the curve as \( t \) varies.

Parametrization is useful because:

• It simplifies the computation and plotting of curves, especially when the shapes are not simple lines.
• It gives insight into how the curve is traversed over time or with change in \( t \).

You can think of \( t \) as a time-like variable that describes how the curve is "drawn," creating a path as \( t \) moves in its interval. Thus, each value of \( t \) determines a specific point along the path, emphasizing the dynamics of the curve's formation.

Plotting Curves

Plotting curves through parametric equations involves translating the equations into points on a graph. This process allows us to visually comprehend the path described by the parametrization. In the exercise, the key points \((-2, 1)\), \((0, 0)\), \((2, 1)\), and \((4, 4)\) are derived by substituting different values of \( t \) into the parametric equations.

Here are the steps for effective plotting:

• Identify strategic \( t \)-values to substitute into your equations to get actual \( (x, y) \) coordinates.


• Plot these coordinates on the graph to establish the curve's major and minor aspects.


• Use the gathered points to sketch the curve, ensuring smooth transitions between them.

With practice, the interpretation of parametric equations through plotting becomes an invaluable skill for visualizing mathematical problems and solutions, helping to see beyond mere numbers and into the forms they create.

Coordinate Plane

The coordinate plane is the two-dimensional world where we graphically represent equations. Comprised of the \( x \)-axis (horizontal) and the \( y \)-axis (vertical), it allows us to map and visualize the behavior of functions and curves.

To properly plot points from parametric equations on this plane:

• Ensure your axes are scaled to include all significant points within the range of your parametrization.


• Plot each point precisely where its \( x \) value corresponds to a horizontal position and its \( y \) value to a vertical position.


• Remember, the intersection point \( (0,0) \) is where both axes meet, giving you a reference for plotting other points.

Understanding how to maneuver and utilize the coordinate plane enhances your ability to graph more complex equations and interpret the information they provide.

Parabolic Path

A parabolic path is a specific type of curve formed by a quadratic equation, as seen in the parametric equations \( x = 2t \) and \( y = t^2 \). In the coordinate plane, this path forms a parabola, a symmetric shape that opens upwards or downwards.

The key characteristics of parabolic paths include:

• Their symmetry about a central axis, often visible as a line of symmetry in the graph.


• The direction of opening, defined by whether the parabola opens upwards or downwards. In mathematical terms, upward or downward orientation is determined by the sign of the coefficient of \( t^2 \).


• Critical points such as the vertex, which in this context is the point where the parabola changes direction, can be discerned from values derived by the parametric equations.

Identifying these elements helps in understanding the nature and structure of quadratic relationships and their graphical representation. Knowing the parabolic path helps predict the curve's behavior and direction, crucial for applications in physics, engineering, and other sciences.