Question

Draw a graph that contains the points $$ (-2,-1),(0,1),(1,3), \text { and }(3,5) $$ Compare your graph with those of other students. Are most of the graphs almost straight lines? How many are "curved"? Discuss the various ways in which these points might be connected.

Short answer

The points should form a straight line, indicating a linear relationship between them.

Step-by-step solution

Plot the Points

Start by plotting the given points on a Cartesian plane. The points to plot are (-2, -1), (0, 1), (1, 3), and (3, 5). Make sure to label each point accordingly.

Draw the Graph

Once the points are plotted, connect them with a line or curve. Try to see if there is a pattern in how the points are arranged.

Analyze the Graph

Look at your graph and identify if the points form a straight line or if they seem to create a curve. A straight line indicates a linear relationship.

Compare with Others

Compare your graph with the graphs of other students. Observe whether their graphs also form straight lines or if some are curved.

Discuss the Connection

Discuss the different ways these points could be connected. For example, using a straight line, which suggests a linear function, or a curve, which suggests a nonlinear relationship.

Key concepts

Cartesian plane

To start, let's understand what a Cartesian plane is. Imagine a sheet of graph paper with a horizontal line and a vertical line crossing at the center. These lines are called axes. The horizontal one is the x-axis, and the vertical one is the y-axis. The point where they meet is the origin, labeled as (0,0).

A Cartesian plane lets you locate points using these axes. Each point is noted as (x, y), showing its position horizontally (x) and vertically (y). For example, the point (-2, -1) means you move 2 units to the left of the origin and 1 unit down. Understanding the Cartesian plane is essential for graphing and analyzing points.

linear relationship

A linear relationship happens when the points you plot form a straight line. This means that as one variable increases, the other also increases at a constant rate, or vice versa. Linear relationships are easy to spot. If you connect the dots and the line is straight, you have a linear relationship.

For example, in the exercise, if the points (-2, -1), (0, 1), (1, 3), and (3, 5) form a straight line when connected, this suggests a linear relationship. Each step in the x-axis results in a consistent step in the y-axis. You can also write a linear relationship as an equation: y = mx + b, where 'm' is the slope, showing the rate of change, and 'b' is the y-intercept, the value of y when x is zero.

nonlinear relationship

Nonlinear relationships are different from linear ones. Here, the points on the graph do not form a straight line. Instead, they might create a curve or some other shape. This suggests that the rate of change between the variables is not constant.

In the exercise, if your graph looks curved or if some sections seem to bend, it indicates a nonlinear relationship. Identifying nonlinear relationships is crucial as they are common in real-world scenarios. For example, the rate of growth in populations, speed, and acceleration in physics all often exhibit nonlinear patterns.

plotting points

Plotting points is a foundational skill in graphing. You start by writing down the given points, like (-2, -1), (0, 1), (1, 3), and (3, 5). Each point has an x-coordinate (horizontal position) and a y-coordinate (vertical position).

First, find the x-coordinate on the x-axis and then the y-coordinate on the y-axis. Plot each point by noting where the x and y coordinates intersect. Label your points for clarity. Accurate plotting is crucial for properly visualizing the data and understanding relationships within the graph.

In the exercise, carefully plotting each of these points will show whether they line up in a straight line (linear) or form a curve (nonlinear).