How does the graph behave at the zeros of f(x)=x(−x+6)(−x−1)2
?(1 point)
Responses
The graph touches the x-axis and turns around at 0 and 6, and touches the x-axis at 1.
The graph touches the x -axis and turns around at 0 and 6, and touches the x -axis at 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 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 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 -1.
The correct response is: The graph crosses the x-axis at 0 and 6, and touches the x-axis and turns around at −1.
To understand how the graph behaves at the zeros of the function f(x), which is given as f(x) = x(-x+6)(-x-1)^2, we need to consider the factors in the expression.
The zeros of the function occur when f(x) is equal to zero. In this case, the function will be zero if any of the factors x, (-x+6), or (-x-1)^2 is equal to zero.
Let's determine the zeros one by one:
1. x = 0: When x is equal to zero, the first factor in the expression becomes zero. This means that the graph will touch the x-axis at x = 0.
2. -x + 6 = 0: This factor represents a linear equation. Solving for x, we have -x + 6 = 0, which gives x = 6. So, the graph will touch the x-axis at x = 6.
3. -x - 1 = 0: This factor represents another linear equation. Solving for x, we have -x - 1 = 0, which gives x = -1. So, the graph will touch the x-axis at x = -1.
So, to summarize the behavior of the graph at the zeros of the function f(x):
- The graph touches the x-axis at x = 0 and x = 6.
- The graph touches the x-axis at x = -1.
Therefore, the correct response is: "The graph touches the x-axis and turns around at 0 and 6, and touches the x-axis at -1."