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Graphing linear equations is an essential skill in understanding the behavior of lines. A linear equation is typically written in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To start graphing, choose two or more values of \(x\) and substitute them into the equation to find the corresponding \(y\) values. For example, for the equation \(y = 2.5x + 4\), choosing \(x = 0\) gives \(y = 4\). When \(x = 2\), \(y = 9\).
These points (0, 4) and (2, 9) provide a straight path to draw the line on the graph. Plotting more points can make the line even more accurate. When you have your points, simply connect them with a straight line.
This method can be used for any linear equation. It's straightforward and ensures you understand the underlying relationship between the variables.

Reflecting a line graph over the x-axis or y-axis is an interesting transformation. When reflecting over the x-axis, keep the x-coordinate unchanged and flip the sign of the y-coordinate. For instance, reflecting the point (0, 4) over the x-axis will give us (0, -4), and (2, 9) becomes (2, -9). Therefore, the new line will be an inversion of the original across the horizontal axis.
This process changes the equation from \(y = 2.5x + 4\) to \(y = -2.5x - 4\). Notice, the slope remains constant while the sign of the whole equation flips.
In the case of reflecting over the y-axis, the procedure is slightly different: switch the sign of the x-coordinate while keeping the y-coordinate the same. For example, reflecting (2, 9) over the y-axis gives us (-2, 9). The new line changes the original equation to \(y = 2.5(-x) + 4\), which simplifies to \(y = -2.5x + 4\). The slope inverts its direction, but the y-intercept remains the same.

Analyzing the equations before and after reflection can deepen our understanding of line transformations. The new equations reveal how the original line's properties are affected. For a reflection over the x-axis, the equation transforms from \(y = 2.5x + 4\) to \(y = -2.5x - 4\). Here, both the slope and intercept signs change. In contrast, a reflection over the y-axis converts the equation to \(y = -2.5x + 4\). Only the slope's sign changes.
These transformations show that reflections over different axes result in distinct changes to the line's equation. When these reflected equations are observed on a graph, they provide visual confirmation. The first reflected line after the x-axis transformation is a mirror image inverted vertically, while the second line is inverted horizontally. This type of equation analysis builds a solid base for understanding fundamental concepts in graph transformations and their algebraic implications.