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Graphing functions involves plotting points on a coordinate plane to visualize the behavior of a mathematical expression. For the function \( y = |4 - x| \), which represents an absolute value function, the plot will typically form a symmetrical "V" shape. The absolute value causes the expression to always be non-negative. This is why, when graphing, you can expect such functions to exhibit a sharp turn or corner at their point of reflection. In our function, this point occurs at \( x = 4 \), where the graph makes a vertex, creating the "V"-shape. To start, select a set of values for \( x \); for instance, values around the point where the function's behavior changes, \( x = 4 \), will be very telling. After calculating the corresponding \( y \) values, you plot these pairs on a graph. This plotted line represents the absolute value function. By connecting these values with a smooth line, you can see that the "V" shape is symmetric around the line \( x = 4 \). The graph gives a visual representation of how \( y \) behaves depending on \( x \) values and aids in understanding the intercepts as well.

The x-intercept is a critical point on a graph where the function's value reaches zero. For the function \( y = |4 - x| \), we determine the x-intercept by setting \( y \) to zero and solving for \( x \). This results in the equation \( 4 - x = 0 \), simplifying to \( x = 4 \). This means the graph crosses the x-axis at this point. At \( x = 4 \), every value that \( y \) takes as \( x \) is altered by this point of reflection. The x-intercept tells you where the function changes direction, which is a unique property of absolute value functions compared to linear or other types of graphs. Thus, knowing the x-intercept allows you to predict symmetry and the nature of the graph on either side of this point.

The y-intercept of a graph is the point where the function crosses the y-axis. This occurs when \( x \) is set to 0. For \( y = |4 - x| \), substituting \( x = 0 \) into the function gives us \( y = |4 - 0| = 4 \). Thus, the y-intercept of this function is at the point \( (0, 4) \). It signifies where the graph touches the y-axis and helps in anchoring the position of the "V" shape on the coordinate plane. The y-intercept provides a starting reference point for plotting a graph and understanding the vertical displacement of the absolute value graph. Since the absolute value function always outputs non-negative values, the y-intercept also indicates that above this point, the function remains positive, reinforcing its non-negative characteristic.

Creating a table of values is an essential step in graphing functions. It involves picking values for \( x \) and computing the corresponding \( y \) values. For \( y = |4 - x| \), a strategic approach is to choose \( x \) values that are centered around the vertex at \( x = 4 \). Here's how it was done:

• For \( x = 2 \), \( y = |4 - 2| = 2 \)
• For \( x = 3 \), \( y = |4 - 3| = 1 \)
• For \( x = 4 \), \( y = |4 - 4| = 0 \)
• For \( x = 5 \), \( y = |4 - 5| = 1 \)
• For \( x = 6 \), \( y = |4 - 6| = 2 \)

With this table, you have a clear path to sketch the graph by plotting these points on a coordinate plane. Observing the pattern can also help in identifying any symmetry or repetitive trends in the function, characterizing the graph. This table serves as a blueprint for connecting these points, offering a visible and tangible guide for graph completion.