Question

he graph of f(x) = |x| is transformed to g(x) = |x + 1| – 7. On which interval is the function decreasing?

(–∞, –7)
(–∞, –1)
(–∞, 1)
(–∞, 7)

Answer 1

To analyze the transformation of the function \( f(x) = |x| \) into \( g(x) = |x + 1| - 7 \), we can follow these steps:

1. **Identify the vertex of \( f(x) = |x| \)**:
The function \( f(x) = |x| \) has a vertex at the point (0, 0).

2. **Transform the graph**:
The transformation \( |x + 1| \) shifts the graph to the left by 1 unit. Thus, the vertex of \( |x + 1| \) moves to (-1, 0).
Then, subtracting 7 shifts the entire graph down by 7 units. So the vertex of \( g(x) = |x + 1| - 7 \) is at the point (-1, -7).

Therefore, we can sketch \( g(x) \). The graph of \( g(x) \) has:
- A vertex at (-1, -7).
- It decreases from \( -\infty \) to -1.
- It increases from -1 to \( +\infty \).

3. **Determine the decreasing interval**:
The function \( g(x) \) is decreasing on the interval \( (-\infty, -1) \).

Considering the provided options, the interval on which the function is decreasing is:

**(–∞, –1)**.