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When we talk about linear functions, we're referring to equations that produce a straight line when graphed on the coordinate plane. A linear function is usually written in the form of the equation \[ f(x) = mx + b \]where:

• \( m \) is the slope of the line. It shows how steep the line is.
• \( b \) is the y-intercept. This is where the line crosses the y-axis.

In our problem, the function is given as \( f(x) = \frac{x-3}{2} \), which can also be thought of as \( f(x) = \frac{1}{2}x - \frac{3}{2} \). Here, \( \frac{1}{2} \) is the slope, showing that for every increase of 1 in \( x \), \( f(x) \) increases by 0.5. The "-3/2" represents where the line would cross the y-axis if extended beyond our limited interval of 0 to 5.

The coordinate plane is like a grid that helps us plot points to visualize equations and functions. It consists of two perpendicular lines:

• The horizontal line, called the x-axis.
• The vertical line, called the y-axis.

These axes split the plane into four quadrants. However, for this problem, we only focus on the positive side of the x-axis, as the exercise's range is from 0 to 5. When you place a point on this grid, it is written like \((x, y)\), where \( x \) is found on the x-axis and \( y \) on the y-axis. The intersection of the x and y axes is known as the origin, which has the coordinates (0,0).Plotting the function involves accurately placing each calculated point from the table of values on this grid. By consistently using the coordinate plane, we can create a clear and precise visualization of any function.

Creating a table of values is a crucial step in graphing functions. This table helps map out which points should be plotted on the coordinate plane. For the function \( f(x) = \frac{x-3}{2} \), we choose several values of \( x \) within the interval 0 to 5 and compute the corresponding \( y \) values.

• Begin by choosing small and simple \( x \) values within the domain provided, such as 0, 1, 2, 3, 4, and 5.
• Plug these numbers into the function to get \( f(x) \), which gives you the y-value.
• Each result creates a coordinate pair \((x, y)\).

For instance, if \( x = 0 \), then \( f(0) = \frac{0-3}{2} = -1.5 \). This makes one pair (0, -1.5). Repeating this operation for each \( x \) within the set range allows you to fill out the table, providing clear coordinates to plot the linear function.

Plotting points on the coordinate plane is like connecting dots to draw a pattern or picture. To begin, you'll need to take each coordinate pair obtained from the table of values, such as (0, -1.5), (1, -1), etc., and mark them clearly.

• The first number of the pair is the x-coordinate, horizontally on the x-axis.
• The second number is the y-coordinate, vertically on the y-axis.

Once all points are plotted, you connect them in sequence. Since the line represents a linear function, you should be able to draw a straight line through these points. Make sure your line only covers the domain specified, which in this case is from \( x = 0 \) to \( x = 5 \). This visual representation helps you confirm that the calculated points fall in line with the expected behavior of the function.