Final answer:
The vertex of the function y = |4x + 8| - 1 is (-2, -1). The range of the function is y ≥ -1, as the lowest point of the V-shaped graph is at y = -1 and the function increases without bound as x moves away from this vertex.
Step-by-step explanation:
The question asks to find the vertex and range of the function y = |4x + 8| − 1.
First, we need to understand that this is an absolute value function, which typically has a V-shaped graph.
The equation inside the absolute value, 4x + 8 = 0, will give us the x-coordinate of the vertex.
Solving for x, we get x = -2. Therefore, the x-coordinate of the vertex is -2.
To find the y-coordinate of the vertex, substitute x = -2 into the function, which yields y = |4(-2) + 8| − 1 = |0| − 1 = 0 − 1 = -1. Thus, the vertex is at (-2, -1).
The range of this function can be found by considering the transformation of the absolute value function by the constant -1.
Since the absolute value function has a minimum value of 0 (when the inside is 0), subtracting 1 from the absolute value function means the minimum value of the entire function y = |4x + 8| − 1 is -1.
As x moves away from -2 in either direction, the function increases without bound.
Therefore, the range is y ≥ -1.