Transformation of Sine Function – Lesson
The composite transformation of the sine function involves transforming the original sine wave by stretching, shifting, and reflecting it in the coordinate plane. The general formula for a composite transformation of the sine function is given by:
\( y=a \sin (b x+c)+d\)
where a affects amplitude, and the amplitude is |a|,
b affects period, the period is \(\frac{2 \pi}{b} \text { for } b>0\)
c affects horizontal translation
d affects vertical translation, the principle axis is \( y=d\)
Combined transformation
\( y=a \sin (b x+c)+d\) obtained from \( sin(x)\)
by a vertical stretch with the scale factor of \( a\)and a horizontal stretch with a scale factor of \( \frac{1}{b},\) units, followed by a horizontal translation of \( c\) units and a vertical translation of \( d\) units.
1. Vertical Stretch: \( y=a \sin x\)
The sine function can be stretched vertically by multiplying the function by a constant. For example, y = a * sin(x) will stretch the sine function by a factor of a, where a is the constant.
2. Horizontal Stretch: \( y = sin (b x)\)
The sine function can be stretched horizontally by multiplying the argument of the function by a constant. For example, y = sin(b x) will stretch the sine function by a factor of b, where k is the constant.
3. Horizontal Shift:\( y=sin ( x-c)\)
The sine function can be shifted horizontally by adding or subtracting a constant to the argument of the function. For example, y = sin(x – c) will shift the sine function c units to the right if c is positive, and c units to the left if c is negative.
4. Vertical Shift:\( y = sin x+d \)
The sine function can be shifted vertically by adding or subtracting a constant to the function. For example, y = sin(x) + d will shift the sine function c units up if d is positive, and d units down if d is negative.
Composite Transormation
The composite transformation of the sine function involves transforming the original sine wave by stretching, shifting, and reflecting it in the coordinate plane. The general formula for a composite transformation of the sine function is given by:
\( y=a \sin (b x+c)+d\)
where a affects amplitude, and the amplitude is |a|,
b affects period, the period is \(\frac{2 \pi}{b} \text { for } b>0\)
c affects horizontal translation
d affects vertical translation, the principle axis is \( y=d\)
Combined transformation