Question

Describe the graph of the function at its roots. f(x) = (x - 2)³ (x + 6)² (x + 12). At x = 2, the graph __ₜhe x-axis. At x= -6, the graph __ₜhe x-axis. At x=-12, the graph __ₜhe x-axis.

a) Crosses, Does not intersect, Touches
b) Touches, Crosses, Does not intersect
c) Does not intersect, Touches, Crosses
d) Crosses, Touches, Does not intersect

Answer 1

The graph of the function crosses the x-axis at x = 2 and x = -12, and touches the x-axis at x = -6, corresponding to answer (d) Crosses, Touches, Crosses.

Step-by-step explanation:

To describe the graph of the function at its roots, we need to consider the behavior of the function at each of the x-intercepts, which are given by the roots of the function. The roots are x = 2, x = -6, and x = -12, and the respective multiplicities of these roots are 3, 2, and 1.

At x = 2, the root has a multiplicity of 3, which is odd. This means that the graph of the function crosses the x-axis at this point.

At x = -6, the root has a multiplicity of 2, which is even. This implies that the graph of the function does not cross the x-axis but rather touches the x-axis and turns back at this point.

Lastly, at x = -12, the root has a multiplicity of 1, which is odd, indicating that the graph crosses the x-axis at this root as well.

The correct descriptions are therefore:

• At x = 2, the graph crosses the x-axis.
• At x= -6, the graph touches the x-axis.
• At x=-12, the graph crosses the x-axis.

So the answer to the student's question would be (d) Crosses, Touches, Crosses.