Question

Graph each set of ordered pairs. Connect them with a curve that seems to you to best fit the data. (0,4),(3,3.2),(5,2),(6,0),(5,-2),(3,-3.2),(0,-4)

Short answer

Plot the points on a graph, then draw a smooth, continuous curve through the points in the given order, checking for symmetry.

Step-by-step solution

Plot the points

Start with a graph paper and plot each of the given ordered pairs on the cartesian plane. Make sure that the horizontal axis (x-axis) and the vertical axis (y-axis) are properly labeled. Plot the points (0,4), (3,3.2), (5,2), (6,0), (5,-2), (3,-3.2), (0,-4) accurately on the graph.

Connect the points

Begin at the first point (0,4) and draw a smooth curve that connects to (3,3.2), (5,2), (6,0), (5,-2), (3,-3.2), and ends at (0,-4). The curve should be continuous and should flow naturally from one point to the next without any sharp turns.

Review the curve

Check the drawn curve to see if it's a best-fit curve that represents the trend of the given data. The curve should be symmetric since the points suggest a pattern that increases and then decreases in a consistent manner, reflecting a possible radial symmetry across the y-axis.

Key concepts

Understanding the Cartesian Plane

The Cartesian plane is a fundamental concept in mathematics, particularly in geometry and algebra. Named after the French philosopher and mathematician René Descartes, it's a two-dimensional surface consisting of two number lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants, each with a distinct combination of positive and negative values.

When graphing ordered pairs like (0,4), (3,3.2), etc., the first number is the x-coordinate indicating the position along the horizontal axis, while the second number is the y-coordinate representing the vertical axis position. For beginners, it's helpful to start with a clearly marked grid and plot each point where the coordinates meet. As you grow comfortable with plotting, you'll be able to visualize and sketch graphs without relying as much on the grid.

Plotting Points with Precision

Plotting points is a critical skill when working with graphs, as it allows us to visually represent relationships and patterns. When given a set of ordered pairs, such as (0,4), (3,3.2), and so on, locating the correct position for each on the Cartesian plane is essential for accuracy. The process starts by identifying the x-coordinate on the horizontal axis and moving vertically to the y-coordinate.

Remember, the intersection of these coordinates is the precise point you need to mark. Using a clear ruler can aid in drawing straight lines to ensure precise plotting. It’s also good practice to label the points with their coordinates, preventing any confusion and making it easier for you to notice any patterns that might emerge from their arrangement.

Drawing the Best-Fit Curve

A best-fit curve is a smooth line that reflects the general pattern of a scatter plot of points, capturing the core tendency of the data. It doesn't need to pass through every point, but rather should strike a balance that minimizes the distance from all points collectively.

After plotting the points, you'll need to visualize how a curve can best represent the overall direction of the data. While drawing, aim for symmetry and smooth transitions, avoiding sharp angles unless the data specifically indicates such a change in direction. Best-fit curves are often used in statistics to represent trends and make predictions based on the graphed data.

Appreciating Radial Symmetry in Graphs

Radial symmetry refers to a pattern that is symmetrical around a center point or axis. In this case, after plotting our ordered pairs and drawing the best-fit curve, we may observe a symmetrical pattern about the y-axis. This implies that for every point on one side of the axis, there is a mirrored point on the opposite side with the same distance from the center.

Recognizing radial symmetry can greatly simplify the task of graphing because it provides a guide for predicting the placement of corresponding points. It also reflects a certain balance in the data, which is important when analyzing the nature of the dataset and can even be aesthetically pleasing, creating a harmonious and balanced graph.