91Ó°ÊÓ

Graphing is a visual way to represent data, equations, or functions on a coordinate plane. It helps us see how variables relate to one another. In the exercise, we're graphing the linear equation \( y = -\frac{1}{2}x \). This equation produces a straight line, which is characteristic of linear equations. Every point on the line satisfies the equation.

To start graphing, we first need to identify and calculate several points by selecting specific values of \(x\). The exercise gives us \(x\) values from -3 to 3, and by plugging these into our equation, we find the corresponding \(y\) values. This creates coordinate pairs or points. For example, when \(x = -3\), \(y = 1.5\), which gives us the point (-3, 1.5).

• Plot each identified point on the graph.
• Use a ruler to connect these points with a straight line.

This line visually represents the equation, showing the continuous relationship between \(x\) and \(y\) as dictated by the equation. The line should extend in both directions across the graph, cutting across it according to the slope of the equation.

The slope of a line is a key characteristic that describes its steepness and direction on a graph. It tells us how much \(y\) changes for a change in \(x\). For the equation \( y = -\frac{1}{2}x \), the slope is \(-\frac{1}{2}\).

The concept of slope is often introduced with the phrase "rise over run," which refers to the vertical change (rise) and corresponding horizontal change (run) between two points on a line. A slope of \(-\frac{1}{2}\) means that for every two units the line moves horizontally (to the right), it drops one unit vertically (downwards). Slope can be positive, negative, zero, or undefined:

• Positive slope: line rises as \(x\) increases.
• Negative slope: line falls as \(x\) increases (as in our equation).
• Zero slope: horizontal line.
• Undefined slope: vertical line.

Understanding the slope helps us predict the behavior of the line across the graph and provide insights into the relationship it represents.

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface on which we can plot points, lines, and curves. It consists of a horizontal axis, known as the \(x\)-axis, and a vertical axis, known as the \(y\)-axis. These axes divide the plane into four quadrants.

Each point on the plane is represented by a pair of numbers, known as coordinates (\(x, y\)). The origin, where the axes intersect, has coordinates (0, 0).

• The horizontal coordinate indicates how far to move left or right from the origin.
• The vertical coordinate shows how far to move up or down.

In the exercise, the coordinate plane is used to plot the points we calculated from the linear equation. By understanding how to navigate this plane, you can effectively graph any equation or set of data, making it a powerful tool for visual representation in mathematics.