Graphing Sine and Cosine Trig Functions

The amplitude is the maximum distance from the midline of the graph to its peak or trough. It is half the distance between the maximum and minimum values of the function.

The period is the length of one complete cycle of the function. For the basic sine and cosine functions, the period is typically $$2\text{pi}$$ . It can be adjusted by the coefficient of $$x$$ in the function.

The general form is $$y = A \times \text{cos}(B(x - C)) + D$$ , where $$A$$ is the amplitude, $$B$$ affects the period, $$C$$ is the horizontal shift, and $$D$$ is the vertical shift.

A positive amplitude indicates that the graph will oscillate above and below the midline, reaching a maximum value of $$D + A$$ and a minimum value of $$D - A$$ .

A horizontal shift moves the graph left or right along the x-axis. It is determined by the value of $$C$$ in the general form of the function.

A vertical shift moves the entire graph up or down along the y-axis, determined by the value of $$D$$ in the general form of the function.

$$y = 3\text{cos}(x - \text{pi}) + 5$$ .

The period is $$\frac{2\text{pi}}{2} = \text{pi}$$ .

Sine functions start at the midline and go up, while cosine functions start at the maximum value. This results in a phase shift of $$\frac{\text{pi}}{2}$$ between them.

The maximum value occurs at the peak of the graph, which is given by $$D + A$$ in the general form of the function.