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Math  /  Trigonometry

Questionf(x)=sin⁡0.1xf(x)=\sin 0.1 x is the following transformation of f(x)=sin⁡xf(x)=\sin x shift or stretch □\square in the xx or yy direction □\square of t,,xt_{,}, x or // □\square number □\square

Studdy Solution

STEP 1

1. The function f(x)=sin⁡0.1x f(x) = \sin 0.1x is a transformation of the basic sine function f(x)=sin⁡x f(x) = \sin x .
2. Transformations can include shifts, stretches, or compressions in either the x x - or y y -direction.
3. The coefficient of x x in the argument of the sine function affects the period of the function.

STEP 2

1. Identify the type of transformation.
2. Determine the direction of the transformation.
3. Determine the nature of the transformation (stretch or compression).
4. Identify the numerical factor involved in the transformation.

STEP 3

Identify the type of transformation by comparing the given function f(x)=sin⁡0.1x f(x) = \sin 0.1x with the standard function f(x)=sin⁡x f(x) = \sin x .
The transformation involves a change in the argument of the sine function from x x to 0.1x 0.1x .

STEP 4

Determine the direction of the transformation.
Since the transformation involves changing the argument of the sine function, it affects the x x -direction.

STEP 5

Determine the nature of the transformation (stretch or compression).
The factor 0.1 0.1 in sin⁡0.1x \sin 0.1x affects the period of the sine function. The period of sin⁡x \sin x is 2π 2\pi , and the period of sin⁡0.1x \sin 0.1x is 2π0.1=20π \frac{2\pi}{0.1} = 20\pi .
Since the period increases, this indicates a horizontal stretch.

STEP 6

Identify the numerical factor involved in the transformation.
The factor 0.1 0.1 is the reciprocal of the stretch factor. Therefore, the stretch factor is 10.1=10 \frac{1}{0.1} = 10 .
The transformation is a stretch in the x x -direction by a factor of 10.
Fill in the blanks: shift or stretch stretch\text{stretch} in the x x or y y direction x\text{x} of t,,x t_{,}, x or / / x\text{x} number 10\text{10}

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