Question

Identify the following key features from the algebraic models for each of the functions given below. Try not to graph the functions. maximum and minimum values; domain and range. f(x)= -2.5 sin(x-35)+1

Answer 1

Step-by-step explanation

We are given the function:

`f(x)=-2.5sin(x-35^0)+1`

We can start with the domain and range of the function:

The function given is a sine function

Domain of a function

`\begin{gathered} -1\le \sin \left(x-35^(\circ \:)\right)\le \:1 \\ -2.5\le \:-2.5\sin \left(x-35^(\circ \:)\right)\le \:2.5 \\ -1.5\le \:-2.5\sin \left(x-35^(\circ \:)\right)+1\le \:3.5 \\ \mathrm{Therefore\:the\:range\:is}:-1.5\le\:f\left(x\right)\le\:3.5 \end{gathered}`

Thus, the range is

`\begin{bmatrix}\mathrm{Solution:}\:&\:-1.5\le \:f\left(x\right)\le \:3.5\:\\ \:\mathrm{Interval\:Notation:}&\:\left[-1.5,\:3.5\right]\end{bmatrix}`

Then we can now get the maximum and minimum values

The critical points are

`\begin{gathered} \mathrm{Plug\:the\:extreme\:point}\:x=-0.95993..+180^(\circ\:)+360^(\circ\:)n \\ \:\mathrm{into}\:-2.5\sin \left(x-35^(\circ \:)\right)+1\quad \Rightarrow \quad \:y=-1.5 \end{gathered}`

So that the minimum will be

`\mathrm{Minimum}\left(-0.95993...+180^(\circ\:)+360^(\circ\:)n,\:-1.5\right)`

Also, the maximum is

`\:\mathrm{Maximum}\left(5.32325...+360^(\circ\:)n,\:3.5\right)`