91Ó°ÊÓ

The concept of a horizontal shift is essential in modifying the position of the basic absolute value function \( g(x) = |x| \). A horizontal shift involves moving the graph left or right along the x-axis.

In our example, the absolute value function is given as \( f(x) = |x+1| - 4 \). The term \( x+1 \) inside the absolute value indicates a horizontal shift. Specifically, a "+1" inside the absolute value suggests that every point on the graph of \( |x| \) moves 1 unit to the left. This shift occurs because replacing \( x \) with \( x+1 \) means that the graph needs to reach zero a step sooner than it did originally. To summarize:

• A "+" value inside \( |x+a| \) shifts the graph \( a \) units to the left.
• A "-" value inside \( |x-a| \) shifts the graph \( a \) units to the right.

Understanding the horizontal shift helps you effectively move the graph along the x-axis, ensuring that you position it correctly for further transformations.

Vertical shift is another fundamental transformation when graphing absolute value functions. This transformation moves the graph up or down the y-axis, changing its vertical position without affecting its shape.

For the function \( f(x) = |x+1| - 4 \), the "-4" at the end of the equation represents a vertical shift. Here, the entire graph of \( |x+1| \) is moved 4 units downwards.

• A "-" value outside \( |x| - b \) represents a downward shift by \( b \) units.
• A "+" value outside \( |x| + b \) would move the graph upwards by \( b \) units.

This vertical shift doesn't alter the direction or orientation of the graph; it simply repositions it along the y-axis. When dealing with transformations, correctly applying vertical shifts helps place the graph appropriately in its final position.

The vertex of an absolute value function is the turning point or the 'V' point where the graph changes direction. For a standard absolute value function \( g(x) = |x| \), the vertex is at the origin, (0,0). However, shifts, both horizontal and vertical, will change the location of the vertex.

In our case, the function is \( f(x) = |x+1| - 4 \). Determining its vertex involves combining the effects of both the horizontal and vertical shifts:

• The function includes a horizontal shift left by 1 unit due to \( x+1 \). Normally, \( g(x) = |x| \) would have its vertex at (0,0), but with this shift, it moves to (-1,0).
• Additionally, the vertical shift of -4 moves the vertex down to (-1, -4).

The new vertex is therefore positioned at (-1, -4). It's important to understand how the vertex reflects the combined effects of these shifts, as it is critical for drawing and analyzing the function's graph.