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Graphing functions is the process of drawing the graph of a mathematical function. For the function \(f(x) = |x - 4| + 1\), graphing begins by understanding the basic shape of the absolute value function \(y = |x|\), which looks like a 'V'. This function is symmetric about the y-axis and has a vertex at the origin (0,0).

To graph \(f(x)\), identify how the function is transformed from the basic \(y = |x|\). Here, it's moved to the right by 4 units because of \(x - 4\), and moved up by 1 unit due to the \(+1\). These transformations alter the basic position of the 'V' shape, shifting its vertex and, thus, its graph.

Once you've identified these transformations, plot the key points and form the 'V' shape accordingly. With the vertex at (4,1), connect it to other key points like (3,0) and (5,0) to finalize the graph, reflecting these transformations accurately.

Vertical transformations change the position of a function's graph along the y-axis. In our function \(f(x) = |x - 4| + 1\), the \(+1\) tells us to lift the graph 1 unit up from its original position. This change affects each point on the graph, moving every y-coordinate up by one.

Vertical shifts are one of the simplest transformations to apply. They don't change the shape of the graph, only its vertical position.

• If you add a number to a function, you shift it upward.
• If you subtract a number, you shift it downward.

Understanding these transformations helps in quickly redrawing familiar functions in new positions on the coordinate plane. For \(f(x)\), this upward shift makes the vertex move to (4,1) from its original lower starting position.

Identifying key points is crucial in graphing functions efficiently. For \(f(x) = |x - 4| + 1\), the most important point to find first is the vertex of the absolute value function, which is influenced by the horizontal shift inside \(|x - h|\). For this function, \(h = 4\), making the initial vertex at (4,0). Then, apply the vertical transformation, which lifts this vertex to (4,1).

Next, find the x-intercepts or any further distinctive points like when \(f(x) = 0\). Solving \(|x - 4| = 0\) provides \(x = 3\) and \(x = 5\), giving points (3,0), (4,1), and (5,0) as crucial points on the graph.

These points help plot the absolute value's V-shape accurately. Knowing how to identify and use these key points simplifies the graphing process and helps represent the function's transformations and shape effectively.