Question

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.

Answer 1

To transform a trigonometric function from the standard trig functions f(x) = sin x, f(x) = cos x, or f(x) = tan x, using key features, you need to consider the amplitude, period, phase shift, vertical shift, and the minimum and maximum points.

Step-by-step explanation:

To transform a trigonometric function from the standard trig functions f(x) = sin x, f(x) = cos x, or f(x) = tan x, using key features, you need to consider the amplitude, period, phase shift, vertical shift, and the minimum and maximum points.

1. Amplitude: The amplitude is the absolute value of the coefficient of the trigonometric function. For example, in f(x) = 2 sin x, the amplitude is 2.

2. Period: The period is the distance between two consecutive maximum points or two consecutive minimum points. It can be calculated as 2π divided by the coefficient of x. For example, in f(x) = sin(2x), the period is π.

3. Phase Shift: The phase shift is the horizontal shift of the graph. It can be calculated as -p/b. For example, in f(x) = sin(x-p/4), the phase shift is p/4 to the right.

4. Vertical Shift: The vertical shift is the vertical translation of the graph. It is determined by adding or subtracting a constant from the function. For example, in f(x) = sin x + 2, the graph is shifted 2 units up.

5. Minimum and Maximum Points: The minimum and maximum points can be determined by considering the amplitude and vertical shift. The maximum point is (0, amplitude + vertical shift), and the minimum point is (0, -amplitude + vertical shift).

Answer 2

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= (2\pi)/(\omega) \\ \\ \therefore \omega= (2\pi)/(T)`

Thus:


`f_(1)(x)=sin((2\pi x)/(T)) \ and \ f_(2)(x)=cos((2\pi x)/(T))`

Regarding the tangent function:


`f_(3)(x)=tan(\omega x)`

where the Period T is given by:


`T= (\pi)/(\omega) \\ \\ \therefore \omega= (\pi)/(T)`

Thus:


`f_(3)(x)=tan((\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:


`(\phi)/(\omega)`

is the phase shift.

4. Vertical shift

The constant
`B` 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=(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((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=(\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((\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((x)/(2)-(\pi)/(3))+4`

So the key features are:


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

So the key features are:


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

So the key features are:


`Amplitude: DNE \\ \\ Period:T= (\pi)/((1)/(2))=\boxed{2\pi} \\ \\ Phase \ shift:\boxed{(\phi)/(\omega)=(2\pi)/(3)} \\ \\ Vertical \ shift:\boxed{2} \\ \\ Maximum \ point: DNE \\ \\ Minimum \ point:DNE`