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Question
What is a graph of the function?

y=|x|−1

(1 point)
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Answer 1

The function \( y = |x| - 1 \) represents a V-shaped graph, which is characterized by the absolute value component. To understand its form, let's break it down:

1. **Absolute Value:** The expression \( |x| \) denotes the absolute value of \( x \). This means it is always non-negative, with output values equal to \( x \) for \( x \geq 0 \) and \( -x \) for \( x < 0 \).

2. **Vertical Shift:** The graph \( y = |x| - 1 \) translates the basic absolute value graph \( y = |x| \) downward by 1 unit.

### Key Features of the Graph:

- **Vertex:** The graph has a vertex (the point at which it changes direction) at \( (0, -1) \).
- **Lines:** For \( x \geq 0 \), the line follows the equation \( y = x - 1 \) (a line with slope 1, starting from \( (0, -1) \)).
- For \( x < 0 \), it follows the equation \( y = -x - 1 \) (a line with slope -1, starting from \( (0, -1) \) as well).

### Summary of Points:
- At \( x = 0 \): \( y = |0| - 1 = -1 \) → Point \( (0, -1) \)
- At \( x = 1 \): \( y = |1| - 1 = 0 \) → Point \( (1, 0) \)
- At \( x = -1 \): \( y = |-1| - 1 = 0 \) → Point \( (-1, 0) \)
- As \( x \) moves away from 0 in either direction, \( y \) increases linearly.

### Visual Representation:
To visualize the graph, you would plot points at \( (-1, 0) \), \( (0, -1) \), and \( (1, 0) \), and then draw straight lines with slopes of +1 and -1 extending outwards from the vertex.

In conclusion, the graph of \( y = |x| - 1 \) is a V-shape with the vertex at \( (0, -1) \), opening upwards.