91Ó°ÊÓ

When graphing transformations, we often manipulate the original graph of a basic function to reflect changes in position, size, or orientation. This is exactly what happens with absolute value functions, such as the given function \( f(x)=|x+2|-8 \).
First, let's understand the basic shape of an absolute value graph, which resembles a "V" shape. The simplest absolute value graph is \( f(x) = |x| \), which is centered at the origin. When a graph undergoes a transformation, it's like taking this basic shape and moving it around the coordinate plane.
Specific transformations occur when:

• Adding or subtracting inside the absolute value affects the horizontal position. Here, \( x+2 \) shifts the graph 2 units left.
• Adding or subtracting outside the absolute value affects the vertical position. Here, \( -8 \) shifts the graph 8 units down.

Understanding these transformations helps to quickly and accurately sketch the graph without needing to re-calculate many points.

The vertex is a key point in graphing absolute value functions. It's where the graph has its minimum (or maximum, if upside-down) point. For the function \( f(x)=|x+2|-8 \), the vertex provides valuable information.
Finding the vertex is straightforward:
- Locate the horizontal shift from \( |x-c| \). In this case, \( x+2 \) means shifting 2 units left, yielding a vertex at \( x = -2 \).
- Calculate the corresponding y-value using the entire function. Substituting \( x = -2 \) into the function gives \( f(-2) = |(-2)+2|-8 = -8 \), so the vertex is \( (-2, -8) \).
The vertex marks the point from which the "V" shape expands outward. In this function, the graph is centered at \( (-2, -8) \), with the arms rising symmetrically as you move left and right from this point.

An absolute value graph characteristically forms a "V" shape, making it easy to identify once familiar with its properties. The standard form \( y = |x| \) provides a symmetric graph centered at the origin, which transforms through the application of horizontal and vertical translations.
For the equation \( f(x)=|x+2|-8 \), the graph centers around the transformed vertex \( (-2, -8) \). Each arm of the "V" extends from this vertex, shaped by the absolute value expression.
To effectively draw this graph:

• Start at the vertex \((-2, -8)\).
• Plot additional points symmetrically, for instance, at \((-3, -7)\) and \((-1, -7)\).

After establishing these key points, the graph's symmetry ensures it branches outward equally on both sides. Observing absolute value forms helps visualize this layout quickly.