Problem 43 - Thomas Calculus

Question

If you have a parametric equation grapher, graph the equations over the given intervals. Parabola \(x=2 t+3, \quad y=t^{2}-1,-2 \leq t \leq 2\)

Step-by-Step Solution

Answer

Plot the points and connect them in a smooth curve to graph the parabola.

Step 1: Understand the Problem

We are given parametric equations: \(x = 2t + 3\) and \(y = t^2 - 1\) with \(-2 \leq t \leq 2\). Our task is to graph these equations over the interval for \(t\). This requires computing \(x\) and \(y\) for various values of \(t\), then plotting these points on a coordinate system.

Step 2: Calculate Points for t

Calculate the corresponding \(x\) and \(y\) for values of \(t\) from \(-2\) to \(2\). Start with \(t = -2\), then \(t = -1\), \(t = 0\), \(t = 1\), and finally \(t = 2\):- For \(t = -2\): \(x = 2(-2) + 3 = -1\), \(y = (-2)^2 - 1 = 3\).- For \(t = -1\): \(x = 2(-1) + 3 = 1\), \(y = (-1)^2 - 1 = 0\).- For \(t = 0\): \(x = 2(0) + 3 = 3\), \(y = 0^2 - 1 = -1\).- For \(t = 1\): \(x = 2(1) + 3 = 5\), \(y = 1^2 - 1 = 0\).- For \(t = 2\): \(x = 2(2) + 3 = 7\), \(y = 2^2 - 1 = 3\).

Step 3: Plot the Points

Plot the coordinates calculated on a graph. The points \((-1, 3)\), \((1, 0)\), \((3, -1)\), \((5, 0)\), and \((7, 3)\) should be plotted on a two-dimensional plane.

Step 4: Draw the Graph

Connect the points with a smooth curve. The curve should represent the path traced by the parametric equations as \(t\) transitions from \(-2\) to \(2\). This will form a parabola.

Key Concepts

Parabola

A parabola is a U-shaped graph of a quadratic function. It can open either upwards or downwards. For our parametric equations, let's explore how it forms. The equation for a parabola in standard form is usually expressed as \( y = ax^2 + bx + c \). In parametric form, however, the equations are split into two, which makes it easier to graph each component separately.

In the given exercise, the parametric form \( x = 2t + 3 \) and \( y = t^2 - 1 \) actually produce the same shape—a parabola. Here, \( x \) is expressed in terms of a parameter \( t \), which translates to the horizontal change on the graph. \( y \) is also expressed as a function of \( t \), reflecting the vertical component. These expressions together describe how the parabola is traced out as the parameter \( t \) changes.

By understanding the parametric equations, one can derive the equation of the parabola. Note that the vertex of this parabola would correspond to the smallest value of \( y \), which is at \( t = 0 \) in this case! Understanding these underlying components helps to comprehend the graph beyond just plotting the points.

Graphing Parametric Curves

Graphing parametric curves is a technique used to represent curves when given parametric equations. In contrast to the traditional method, which involves a single equation, parametric equations involve two separate equations, one for each coordinate, which are dependent on a parameter (often denoted as \( t \)).

The main advantage of parametric equations is that they allow the representation of complex curves and motions, which might otherwise be difficult to express. In the given problem, choosing different values of \( t \) between \(-2\) and \(2\) lets us graph distinct points where the path of the curve can be seen. The process consists of:

• Picking a range for \( t \) (here from \(-2\) to \(2\))
• Calculating the corresponding \( x \) and \( y \) values for each chosen \( t \)
• Plotting these \( (x, y) \) points on a grid
• Connecting them smoothly to reveal the curve

Being comfortable with parametric curves helps in understanding various phenomena like the path of a celestial body or the trajectory of a projectile. Once the plotted points are connected, they will accurately represent the curve's signature pattern, like a parabola shown in the exercise.

Coordinate Plotting

Coordinate plotting is a fundamental skill in graphing where you plot points on a plane using coordinates \(x, y\). Using parametric equations, each stock \( t \) value gives us a unique \(x\) and \(y\) setting on the coordinate plane.

For instance, in the exercise provided, using different values of \( t \), such as \( t = -2 \), \( -1 \), \( 0 \), \( 1 \), and \( 2 \), results in different \( (x, y) \) pairs. Those pairs, determined by the calculations \( (x, y) = (-1,3), (1,0), (3,-1), (5,0), (7,3) \), are plotted on a two-dimensional graph.

The process involves:

• Identifying each pair of calculated coordinates
• Marking each point accurately according to its \(x\) and \(y\) values
• Double-checking to ensure they align with the intended curve

The plotting gives a visual representation of how the equations change over a defined interval, which is essential for interpreting and understanding various mathematical or real-world scenarios. Highlighting these aspects simplifies visualization and helps connect the dots from abstract equations to tangible graphs. By doing so, students can better grasp mathematical concepts through visual learning.