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A table of values is a simple yet powerful tool to help us understand how equations relate to graphs. When dealing with linear equations, creating a table lets us input various values for one variable, usually 'x'. These values are then used to find the corresponding 'y' values based on the given equation.

For example, consider the equation \(y = -x + 4\). To make a table of values, we choose several 'x' values, such as -2, 0, and 2. Substituting these into our equation gives us the corresponding 'y' values as follows:

• When \(x = -2\), \(y = 6\)
• When \(x = 0\), \(y = 4\)
• When \(x = 2\), \(y = 2\)

Thus, our table of values comprises the pairs \((-2,6), (0,4), (2,2)\). This method helps in graphing as it provides clear coordinate points to plot.

The coordinate plane is a crucial tool in mathematics for representing points graphically. It consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).

The intersection point of these axes is called the origin, denoted as \((0,0)\). Together, they form a two-dimensional space where we can plot points such as those from our table of values.

Plotting points is executed by placing a dot where the x-value on the horizontal axis and the y-value on the vertical axis meet. For the equation \(y=-x+4\), we plot the points \((-2, 6), (0, 4), (2, 2)\). This visual representation is straightforward and essential for understanding the relationship between variables.

Linear functions describe relationships where the rate of change between two variables remains constant. These functions create straight lines when graphed. A standard form of a linear equation is \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept that indicates where the line crosses the y-axis.

In the equation \(y = -x + 4\), the slope \(m=-1\), meaning that as \(x\) increases by 1, \(y\) decreases by 1. The y-intercept \(b=4\) tells us that the line crosses the y-axis at \((0, 4)\).

• Understanding the slope helps explain the direction and steepness of the line.
• The y-intercept provides a starting point for plotting the graph.

This foundational understanding of linear functions is vital for interpreting and graphing equations effectively.