Asked by SS
How does the graph behave at the zeros of f(x)=x(−x+6)(−x−1)2
?(1 point)
Responses
The graph crosses the x-axis at 0 and −6
, and touches the x-axis and turns around at 1.
The graph crosses the x -axis at 0 and negative 6 , and touches the x -axis and turns around at 1.
The graph crosses the x-axis at 0 and 6, and touches the x-axis and turns around at −1
.
The graph crosses the x -axis at 0 and 6, and touches the x -axis and turns around at negative 1 .
The graph crosses the x-axis at 6, and touches the x-axis and turns around at −1
.
The graph crosses the x -axis at 6, and touches the x -axis and turns around at negative 1 .
The graph touches the x-axis and turns around at 0 and 6, and touches the x-axis at 1.
?(1 point)
Responses
The graph crosses the x-axis at 0 and −6
, and touches the x-axis and turns around at 1.
The graph crosses the x -axis at 0 and negative 6 , and touches the x -axis and turns around at 1.
The graph crosses the x-axis at 0 and 6, and touches the x-axis and turns around at −1
.
The graph crosses the x -axis at 0 and 6, and touches the x -axis and turns around at negative 1 .
The graph crosses the x-axis at 6, and touches the x-axis and turns around at −1
.
The graph crosses the x -axis at 6, and touches the x -axis and turns around at negative 1 .
The graph touches the x-axis and turns around at 0 and 6, and touches the x-axis at 1.
Answers
Answered by
GPT-4o mini
To analyze the behavior of the graph of the function \( f(x) = x(-x + 6)(-x - 1)^2 \) at its zeros, we first need to find the zeros and the nature of each zero:
1. **Finding the Zeros**:
- The term \( x = 0 \) gives a zero at \( x = 0 \).
- The term \( -x + 6 = 0 \) gives \( x = 6 \).
- The term \( (-x - 1)^2 = 0 \) gives a zero at \( x = -1 \) with a multiplicity of 2 (since it is squared).
So, the zeros of the function \( f(x) \) are at \( x = 0 \), \( x = 6 \), and \( x = -1 \).
2. **Nature of the Zeros**:
- The zero at \( x = 0 \) has multiplicity 1, which means the graph will **cross** the x-axis at this point.
- The zero at \( x = 6 \) also has multiplicity 1, meaning the graph will **cross** the x-axis at this point as well.
- The zero at \( x = -1 \) has multiplicity 2, which means the graph will **touch** the x-axis and turn around at this point.
Now, we can summarize the behavior:
- The graph **crosses** the x-axis at \( x = 0 \).
- The graph **crosses** the x-axis at \( x = 6 \).
- The graph **touches** the x-axis at \( x = -1 \) and turns around.
Given this information, the correct response is:
**The graph crosses the x-axis at 0 and 6, and touches the x-axis and turns around at -1.**
1. **Finding the Zeros**:
- The term \( x = 0 \) gives a zero at \( x = 0 \).
- The term \( -x + 6 = 0 \) gives \( x = 6 \).
- The term \( (-x - 1)^2 = 0 \) gives a zero at \( x = -1 \) with a multiplicity of 2 (since it is squared).
So, the zeros of the function \( f(x) \) are at \( x = 0 \), \( x = 6 \), and \( x = -1 \).
2. **Nature of the Zeros**:
- The zero at \( x = 0 \) has multiplicity 1, which means the graph will **cross** the x-axis at this point.
- The zero at \( x = 6 \) also has multiplicity 1, meaning the graph will **cross** the x-axis at this point as well.
- The zero at \( x = -1 \) has multiplicity 2, which means the graph will **touch** the x-axis and turn around at this point.
Now, we can summarize the behavior:
- The graph **crosses** the x-axis at \( x = 0 \).
- The graph **crosses** the x-axis at \( x = 6 \).
- The graph **touches** the x-axis at \( x = -1 \) and turns around.
Given this information, the correct response is:
**The graph crosses the x-axis at 0 and 6, and touches the x-axis and turns around at -1.**
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