Final answer:
The vertex form of an absolute value equation is V = a|f - h| + k, which includes transformations such as vertical flips, stretches or shrinks, and horizontal and vertical shifts determined by a, h, and k.
Step-by-step explanation:
The vertex form of an absolute value equation is typically written as V = a|f - h| + k, where 'V' is the value of the vertical axis, 'a' represents the slope of the function, 'f' represents the horizontal axis variable, 'h' is the horizontal shift (or the x-coordinate of the vertex), and 'k' is the vertical shift (or the y-coordinate of the vertex). The absolute value function makes a 'V' shape, and the vertex is the point where the graph changes direction.
For the transformations:
If a > 0, the graph opens upwards, and if a < 0, it opens downwards.
The absolute value of 'a' indicates how 'steep' or 'wide' the graph is. Larger values make the graph narrower, smaller values make it wider.
The 'h' value moves the graph left or right. If 'h' is positive, the graph shifts to the right, and if negative, to the left.
The 'k' value represents the vertical shift. Positive 'k' moves the graph up, while negative 'k' moves it down.
As a simple example, let's write the equation V = 2|f - 3| + 4. Here, 'a' is 2, so the graph opens upward and is relatively narrow. The graph is shifted 3 units to the right (since h=3) and 4 units upwards (since k=4).