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 answer

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 \( x = -1 \).

  1. 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.