Two Parent Sinusoidal Functions

The sine function and the cosine function are referred to as sinusoidal functions.

Their graphs have the property that they oscillate above and below a central horizontal line.

For both \(y = \sin(x)\) and \(y = \cos(x)\), this central horizontal line or axis is \(y = 0\).

The equation of the horizontal axis is \(y = \dfrac{\text{maximum value} + \text{ minimum value}}{2}\).

The sinusoidal functions are cyclic. That is, the function values repeat over regular intervals of the domain.

We say that these functions are periodic.

The horizontal length of one cycle is called the period.

The period of both \(y = \sin(x)\) and \(y = \cos(x)\) is \(2\pi\) radians or \(360^{\circ}\).

The amplitude is the perpendicular distance from the central horizontal axis to either a maximum or minimum point on the curve.

We can calculate the amplitude with the formula

amplitude = \(\dfrac{\text{maximum value} - \text{ minimum value}}{2}\)

For both functions, \(y = \sin(x)\) and \(y = \cos(x)\):

The domain is \(\left\{ x \mid x \in \mathbb{R} \right\}\) and

the range is \(\left\{ y \mid -1 \leq y \leq 1, y \in \mathbb{R} \right\}\).

We will refer to \(y = \sin(x)\) and \(y = \cos(x)\) as the parent sinusoidal functions.

For \(y=\sin(x)\), we identified five key points in one period starting on the \(y\)-axis.

In radian measure, these points are \((0,0), \left( \dfrac{\pi}{2} , 1 \right) , (\pi, 0), \left( \dfrac{3\pi}{2} , -1 \right)\), and \((2\pi , 0)\).

In degrees, the five key points are \((0,0), (90^{\circ} , 1), (180^{\circ} , 0), (270^{\circ} , -1)\), and \((360^{\circ} , 0)\).

Once a five-point sketch is complete, we can sketch as many periods as we require.

For \(y = \cos(x)\), we identified five key points in one period starting on the \(y\)-axis.

In radian measure, these points are \((0,1), \left( \dfrac{\pi}{2} , 0 \right) , (\pi , -1) , \left( \dfrac{3\pi}{2} , 0 \right)\), and \((2\pi , 1)\).

In degrees, the five key points are \((0,1) , (90^{\circ} , 0) , (180^{\circ} , -1) , (270^{\circ} , 0)\), and \((360^{\circ} , 1)\).

Once the five point sketch is complete, we can sketch as many periods as we require.

In an earlier module, we looked at transformations. Transformations on a function \(y = f(x)\) can be identified when the function is written in the form \(y = {\color{BrickRed}a}f[{\color{NavyBlue}b}(x-{\color{Mulberry}h})]+{\color{Violet}k}\).

The Sine Function

\(y = {\color{BrickRed}a} \sin[{\color{NavyBlue}b}(x-{\color{Mulberry}h})]+{\color{Violet}k}\)

The Cosine Function

\(y = {\color{BrickRed}a} \cos[{\color{NavyBlue}b}(x-{\color{Mulberry}h})]+{\color{Violet}k}\)

We will review the role of the parameters \({\color{BrickRed}a}, {\color{NavyBlue}b}, {\color{Mulberry}h}\) and \({\color{Violet}k}\) in transforming the sinusoidal functions.

But first, use the Maple Worksheet to look at the effect of changing the values of \({\color{BrickRed}a}, {\color{NavyBlue}b}, {\color{Mulberry}h}\) and \({\color{Violet}k}\).