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After adding the parameters |a,| |b,| |h,| and |k| to the basic function |f(x)=\sin(x),| we obtain the standard form (also called transformed form) of the sine function.
The standard form of a sine function is:
||f(x)=a\sin\big(b(x-h)\big)+k|| where |a|, |b|, |h,| and |k| are real numbers that function as parameters.
Note: The parameters |a| and |b| are always non-zero.
Experiment with the parameters |a,| |b,| |h,| and |k| in the interactive animation to see their effects on the sine function. Observe the changes that take place on the transformed curve (in green) compared to the basic function (in black). Afterwards, keep reading the concept sheet to understand each of the parameters.
The amplitude of a sine function equals half of the distance between the maximum and the minimum values of the function.
Aside from changing the vertical scaling, the parameter |a| is also responsible for the graph’s orientation. As with the majority of functions with the parameter |a|, the sine function is reflected across the |x|-axis when |a| is negative.
To find the value of the amplitude, use the maximum and minimum of the sine function that you are working with.
|\text{Amplitude} = \dfrac{\max - \min}{2}= {\mid}a{\mid}|
In other words, the larger |\mid a \mid| is, the greater the amplitude of the sine function and the more it is stretched vertically, and vice versa.
The period is the distance between two consecutive maxima or minima on the function.
The parameter |b| is responsible for a horizontal scaling by |\dfrac{1}{b}.|
Just like the parameter |a,| the parameter |b| can also change the orientation of the graph. When |b| is negative, the function is reflected across the |y|-axis.
|{\mid}b{\mid} = \displaystyle \frac{2\pi}{\text{period}}|
In summary, the greater the value of the period, the greater the distance between two maxima or between two minima of the function, and vice versa.
On the other hand, an increased period will decrease the value of the parameter |b.|
The phase shift is the horizontal displacement of the function’s |y|-intercept |(0,0)|. It is represented by the letter |h| in a sine function in standard form.
When the parameter |h| is positive, the graph of the sine function moves to the right.
When the value of |h| is negative, the graph of the sine function moves to the left.
This is the vertical displacement of the |y|-intercept |(0, 0)|. It is denoted by the letter |k| in a sine function in standard form.
To determine the value of |k|, use the maximum and minimum value of the function you are working with.
|\displaystyle k = y_{avg} = \frac{\max f+\min f}{2}|
When the parameter |k| has a positive value, the graph of the sine function moves upwards. On the other hand, if the value of the parameter |k| is negative, the graph moves downwards.
An inflection point of a sinusoidal function is a point where the curve changes concavity.
A sine function has an infinite number of inflection points. Each of them are located on the curve halfway between a maximum and a minimum.
|\text{Inflection point}=(h,k)|
|\text{Inflection points}=\left(h+\dfrac{n\text{P}}{2},\ k\right)\quad \text{where }n \in \mathbb{Z}\ \text{and P is the period}|