Question
Which graph represents the function r(x) = |x – 2| – 1
On a coordinate plane, an absolute value graph has a vertex at (2, negative 1).
On a coordinate plane, an absolute value graph has a vertex at (negative 2, negative 1).
On a coordinate plane, an absolute value graph has a vertex at (1, negative 2).
On a coordinate plane, an absolute value graph has a vertex at (negative 1, negative 2).
On a coordinate plane, an absolute value graph has a vertex at (2, negative 1).
On a coordinate plane, an absolute value graph has a vertex at (negative 2, negative 1).
On a coordinate plane, an absolute value graph has a vertex at (1, negative 2).
On a coordinate plane, an absolute value graph has a vertex at (negative 1, negative 2).
Answers
GPT-4o mini
To determine which graph represents the function \( r(x) = |x - 2| - 1 \), we need to analyze the function and identify its properties, especially the vertex.
The function \( r(x) = |x - 2| - 1 \) is an absolute value function. The general form of an absolute value function can be expressed as \( r(x) = |x - h| + k \), where \( (h, k) \) is the vertex of the v-shaped graph.
In the given function:
- The term inside the absolute value, \( x - 2 \), indicates that the vertex's x-coordinate is \( h = 2 \).
- The term outside the absolute value, which is \(-1\), indicates that the vertex's y-coordinate is \( k = -1 \).
Therefore, the vertex of the graph of \( r(x) \) is at the point \( (2, -1) \).
Now, we can look at the given options:
1. **(2, -1)** - This corresponds to the vertex we calculated.
2. **(-2, -1)** - This does not match.
3. **(1, -2)** - This does not match.
4. **(-1, -2)** - This does not match.
The graph that represents the function \( r(x) = |x - 2| - 1 \) is the one with the vertex at **(2, -1)**.
The function \( r(x) = |x - 2| - 1 \) is an absolute value function. The general form of an absolute value function can be expressed as \( r(x) = |x - h| + k \), where \( (h, k) \) is the vertex of the v-shaped graph.
In the given function:
- The term inside the absolute value, \( x - 2 \), indicates that the vertex's x-coordinate is \( h = 2 \).
- The term outside the absolute value, which is \(-1\), indicates that the vertex's y-coordinate is \( k = -1 \).
Therefore, the vertex of the graph of \( r(x) \) is at the point \( (2, -1) \).
Now, we can look at the given options:
1. **(2, -1)** - This corresponds to the vertex we calculated.
2. **(-2, -1)** - This does not match.
3. **(1, -2)** - This does not match.
4. **(-1, -2)** - This does not match.
The graph that represents the function \( r(x) = |x - 2| - 1 \) is the one with the vertex at **(2, -1)**.