Mathematics, 13.12.2019 09:31 nothingworksoutforme

Describe how to transform a trigonometric function from the standard trig function f(x) = sin x, f(x) = cos x, or f(x) = tan x using key features. include amplitude, period, phase shift, vertical shift, the minimum point and a maximum point.

Answers: 1

Show answers

Answers

Answer from: ctdavis

We have the following functions:

$f_{1}(x)=sin(x) \\ f_{2}(x)=cos(x) \\ f_{3}(x)=tan(x)$

So let's analyze how to transform these functions by using key features.

1. Including amplitude.

To include the amplitude in a sine or cosine functions we simply multiply the function by A, so:

$f_{1}(x)=Asin(x) \ or \\ f_{2}(x)=Acos(x)$

The absolute value of A is the amplitude. that is:

$Amplitude=\left | A \right |$

The tangent function have no amplitude because this function grows without a bound.

2. Including period.

Let $\omega$ be a positive real number. The period of the sine or cosine functions is obtained as follows:

$f_{1}(x)=sin(\omega x) \ and \ f_{2}(x)=cos(\omega x)$

The period T is given by:

$T= \frac{2\pi}{\omega} \\ \\ \therefore \omega= \frac{2\pi}{T}$

Thus:

$f_{1}(x)=sin(\frac{2\pi x}{T}) \ and \ f_{2}(x)=cos(\frac{2\pi x}{T})$

Regarding the tangent function:

$f_{3}(x)=tan(\omega x)$

where the Period T is given by:

$T= \frac{\pi}{\omega} \\ \\ \therefore \omega= \frac{\pi}{T}$

Thus:

$f_{3}(x)=tan(\frac{\pi x}{T})$

3. Phase shift

The constant $\phi$ in the equations:

$f_{1}(x)=sin(\omega x\pm \phi) \\ f_{2}(x)=cos(\omega x\pm \phi) \\ f_{3}(x)=tan(\omega x\pm \phi)$

creates a horizontal translation (shift) of the basic functions. To the left (if positive) or to the right (if negative). The number:

$\frac{\phi}{\omega}$

is the phase shift.

4. Vertical shift

The constant

in the equations:

$f_{1}(x)=sin(x)+B \\ f_{2}(x)=cos(x)+B \\ f_{3}(x)=tan(x)+B$

creates a vertical translation (shift) of the basic functions. Upward (if positive) or downward (if negative)

5. Minimum point.

To find the minimum point in a sine or cosine functions let's take the functions:

$f_{1}(x)=Asin(x) \ or \\ f_{2}(x)=Acos(x)$

The minimum point in a sine function is given when this function is minimum, that is, when $x=\frac{3\pi}{2}$ . On the other hand, the minimum point of the cosine function is given when $x=\pi$ , then the minimum points are:

$Sine \ function: \\ \\ min(\frac{3\pi}{2},-A) \\ \\ \\ Cosine \ function: \\ \\ min(\pi,-A)$

There is no a minimum point of the tangent function because it grows without a bound.

6. Maximum point.

To find the Maximum point in a sine or cosine functions let's take the functions:

$f_{1}(x)=Asin(x) \ or \\ f_{2}(x)=Acos(x)$

So the Maximum point of the sine function is given when this function is Maximum, that is, when $x=\frac{\pi}{2}$ . On the other hand, the maximum point of the cosine function is given when x=0

, then the maximum points are:

$Sine \ function: \\ \\ Max(\frac{\pi}{2},A) \\ \\ \\ Cosine \ function: \\ \\ Max(0,A)$

There is no a maximum point in a tangent function because it grows negatively without a bound.

7. Example for the sine function.

As shown in Figure 1 we have the graph of the following function:

$f_{1}(x)=3sin(\frac{x}{2}-\frac{\pi}{3})+4$

So the key features are:

$Amplitude: \boxed{3} \\ \\ Period:T= \frac{2\pi}{\frac{1}{2}}=\boxed{4\pi} \\ \\ Phase \ shift:\boxed{\frac{\phi}{\omega}=\frac{2\pi}{3}} \\ \\ Vertical \ shift:\boxed{4} \\ \\ Maximum \ point: when \ \frac{x}{2}- \frac{\pi}{3}= \frac{\pi}{2} \therefore x= \frac{5\pi}{3} \ then \ \boxed{Max(\frac{5\pi}{3},7)} \\ \\ Minimum \ point:when \ \frac{x}{2}- \frac{\pi}{3}= \frac{3\pi}{2} \therefore x= \frac{11\pi}{3} \ then \ \boxed{min(\frac{11\pi}{3},1)}$

8. Example for the cosine function.

As shown in Figure 2 we have the graph of the following function:

$f_{2}(x)=5cos(\frac{x}{2}+\frac{\pi}{3})+2$

So the key features are:

$Amplitude: \boxed{5} \\ \\ Period:T= \frac{2\pi}{\frac{1}{2}}=\boxed{4\pi} \\ \\ Phase \ shift:\boxed{ \frac{\phi}{\omega}=\frac{2\pi}{3}} \\ \\ Vertical \ shift:\boxed{2} \\ \\ Maximum \ point: when \ \frac{x}{2}+\frac{\pi}{3}=0 \therefore x= \frac{-2\pi}{3} \ then \ \boxed{Max(-\frac{2\pi}{3},7)} \\ \\ Minimum \ point:when \ \frac{x}{2}+ \frac{\pi}{3}= -\pi \therefore x=-\frac{8\pi}{3} \\ or \ x=-\frac{8\pi}{3}+4\pi=\frac{4\pi}{3} \then \ \boxed{min(\frac{4\pi}{3},-3)}$

9. Example for the tangent function.

As shown in Figure 3 we have the graph of the following function:

$f_{2}(x)=5tan(\frac{\pi x}{2}+\frac{\pi}{3})+2$

So the key features are:

$Amplitude: DNE \\ \\ Period:T= \frac{\pi}{\frac{1}{2}}=\boxed{2\pi} \\ \\ Phase \ shift:\boxed{\frac{\phi}{\omega}=\frac{2\pi}{3}} \\ \\ Vertical \ shift:\boxed{2} \\ \\ Maximum \ point: DNE \\ \\ Minimum \ point:DNE$

Describe how to transform a trigonometric function from the standard trig function f(x) = sin x, f(x

Describe how to transform a trigonometric function from the standard trig function f(x) = sin x, f(x

Answer from: Quest

points are handed out whenever you post answers.

brainliest answers are handed out by the person asking the question, so whether your answer is picked depends entirely on whether the asker bothers to pick

if you're asking about your rank, it depends on having a certain number of points and a certain number of brainliest answers. for example, at the time of this writing, i have 44338 points out of 15000 needed for the next (nearly triple the requirement! ) but i only have 24 out of 50 required brainliest answers. this number decays, meaning after a certain amount of time (i think about a month), the total number of brainliest answers you've earned doesn't count toward this total. (i've been stuck at the same rank for a long

the lesson to learn here is that you shouldn't worry about points or rank. chances are, no matter how good an answer you may provide, your efforts will go mostly unnoticed unless you focus on the newest/most active questions that get posted.

Answer from: Quest

Just a question- could you organize the data better? i just cannot tell which number are for which bakery. i will be happy to answer this question if you could do this for me! : )

Another question on Mathematics

Mathematics, 21.06.2019 12:30

Elijah and his sister went to the movies. they had $34 altogether and spent $9.50 per ticket. elijah and his sister bought the same snacks. write and solve an inequality fotlr the amount that each person spent on snacks. interpret the solition

Answers: 3

Answer

Mathematics, 21.06.2019 15:40

Which point on the x-axis lies on the line that passes through point c and is parallel to lino ab?

Answers: 1

Answer

Mathematics, 21.06.2019 16:30

Marco is studying a type of mold that grows at a fast rate. he created the function f(x) = 345(1.30)x to model the number of mold spores per week. what does the 1.30 represent? how many mold spores are there after 4 weeks? round your answer to the nearest whole number. 1.30 represents the number of mold spores marco had at the beginning of the study; 985 mold spores 1.30 represents the number of mold spores grown in a given week; 448 mold spores 1.30 represents the 30% rate of weekly growth; 345 mold spores 1.30 represents the 30% rate of weekly growth; 985 mold spores

Answers: 1

Answer

Mathematics, 21.06.2019 22:30

What is the point slope form of a line with slope -5 that contains point (2,-1)

Answers: 1

Answer

You know the right answer?

Describe how to transform a trigonometric function from the standard trig function f(x) = sin x, f(x...

Questions

Mathematics, 26.03.2021 21:50

• The rest of the students fill 6 buses. Enter an equation that can be solved to find the number of students, s, that are on each bus. Enter your res...

Mathematics, 26.03.2021 21:50