Question

Sketch the graph of the given parametric equations by hand, making a table of points to plot. Be sure to indicate the orientation of the graph. \(x=t^{3}-t+3, \quad y=t^{2}+1, \quad-2 \leq t \leq 2\)

Short answer

1. Create a table of values for \( t \), \( x \), and \( y \) for \(-2 \leq t \leq 2\). 2. Plot points, sketch the curve, and indicate upward arrows from -2 to 2.

Step-by-step solution

Understand the Parametric Equations

The given parametric equations are:\[ x = t^3 - t + 3 \] and \[ y = t^2 + 1 \]. These equations describe how the x and y coordinates change with the parameter \( t \). We need to determine how \( x \) and \( y \) change as \( t \) varies from -2 to 2.

Create a Table of Values

We will create a table of values for \( t \), \( x \), and \( y \) by choosing several values for \( t \) within the given range \(-2 \leq t \leq 2\). It is common to choose integer values for ease of calculation. The table will help us to sketch the curve.| \( t \) | \( x = t^3 - t + 3 \) | \( y = t^2 + 1 \) || ------ | ------------------ | ---------------- || -2 | \((-2)^3 - (-2) + 3 = -8 + 2 + 3 = -3\) | \((-2)^2 + 1 = 4 + 1 = 5\) || -1 | \((-1)^3 - (-1) + 3 = -1 + 1 + 3 = 3\) | \((-1)^2 + 1 = 1 + 1 = 2\) || 0 | \((0)^3 - (0) + 3 = 3\) | \((0)^2 + 1 = 1\) || 1 | \(1^3 - 1 + 3 = 3\) | \(1^2 + 1 = 2\) || 2 | \(2^3 - 2 + 3 = 8 - 2 + 3 = 9\) | \(2^2 + 1 = 5\) |

Plot the Points

Using the table from Step 2, plot the points \((-3, 5)\), \((3, 2)\), \((3, 1)\), \((3, 2)\), and \((9, 5)\) on a coordinate plane. Mark each point clearly and determine the orientation by examining how \( t \) progresses from -2 to 2.

Sketch the Curve and Indicate the Orientation

Connect the plotted points smoothly to form the curve of the parametric equations. Observe how \( x \) and \( y \) change as \( t \) moves from -2 to 2 and draw arrows along the curve to indicate the direction of increasing \( t \). Notice that the orientation goes from left to right as \( t \) increases in this case.

Key concepts

Coordinate Plane

The coordinate plane is a foundational concept in graphing equations, including parametric equations. It is a two-dimensional surface on which we can plot points defined by pairs of numbers. Each pair is known as

• (x, y), where x is the horizontal coordinate
• and y is the vertical coordinate.

This grid-like structure helps us understand the spatial relationship between points.

When dealing with parametric equations, such as the ones given in the exercise, each value of the parameter \( t \) corresponds to a specific point,

• (x, y) on the coordinate plane.

This makes it easier to visualize how the curve forms as \( t \) changes.

The coordinate plane is divided by the x-axis and y-axis, which intersect at the origin

• (0, 0).

Understanding these basic elements helps you plot points accurately, facilitating the curve sketching process.

Curve Sketching

Curve sketching is the process of drawing the overall shape of a graph based on plotted points from the parametric equations. It's a skill that allows you to visualize the path represented by the parametric equations.

With the given functions

• \( x = t^3 - t + 3 \)
• and \( y = t^2 + 1 \),

you need to find corresponding x and y values for each t value and plot these on the coordinate plane.

To effectively sketch the curve:

• Select appropriate values for \( t \), often integers within the specific range, like
  • -2, -1, 0, 1, and 2 in our example.
• Use these values to calculate x and y, creating a table of points.
• Plot these points on the coordinate plane.
• Connect the points with a smooth line, drawing the graph.

Curve sketching provides a powerful graphical representation of how the parameters affect the x and y positions across the range of \( t \).

Orientation of Graph

Understanding the orientation of a graph is essential when working with parametric equations. Unlike regular functions, parametric equations can describe curves that re-trace parts of their path or change direction in unique ways.

Orientation refers to the path a graph takes as \( t \) increases. In parametric plotting, the direction is not always obvious. Thus, it's crucial to indicate how the graph progresses. Draw arrows along the curve to show which direction it moves as \( t \) increases from its minimum to maximum.

In our exercise, as \( t \) progresses from -2 to 2:

• The graph starts at \((-3, 5)\),
• moves through (3, 2), (3, 1) repeatedly,
• and finally reaches \((9, 5)\).

Marking arrows along the path reveals the orientation direction, which in this scenario, the graph moves from left to right.

This concept helps in understanding any cyclical movements or revisits within the curve—that are common in parametric systems.