Question

Write the equation for these:

Answer 1

1. The equation for the graph for the graph provided is
`(y = -\sin(x))`.

2. The equation for the cosecant function satisfying the given conditions: period of
`(5\pi)` and range of
`((- \infty, -6] \cup [6, \infty))` can be written as
`[y = -6\csc\left((2\pi)/(5\pi)x + \phi\right) + k,]`.

An equation is a mathematical function that depicts the equality or equivalence between mathematical expressions.

1) The equation for the graph is a sine function that has been reflected over the x-axis.

The curve starts from the origin (0,0), goes up to touch y = 4, then descends passing through the origin again, and continues downwards to touch y = -4 before ascending again.

Two specific points on the x-axis are labeled:
`((2\pi))/(7)` and
`((4\pi))/(7)`.

2) For the cosecant function,

`[y = -6\csc\left((2\pi)/(5\pi)x + \phi\right) + k,]`

(k) is the vertical shift and
`(\phi)` is the phase shift. Please note that the values of (k) and
`(\phi)` would depend on the specific graph of the function.

Thus, the equations for the graph and the cosecant function have been written as above.