Trig Graphs and Transformations

Trigonometric Graphs

The sine, cosine, and tangent functions each have unique characteristics when plotted as graphs.
The sin(θ) graph starts at the origin, peaks at 1, drops to -1, and then returns to the origin, completing a cycle. This pattern continues indefinitely. The period of the sin function is 2π.
The cos(θ) graph starts at 1, drops to -1, and then returns to 1, completing a cycle. This pattern also continues indefinitely. The period of the cos function is 2π just like the sin function.
The tan(θ) graph rises from negative infinity to positive infinity at π/2 and has vertical lines drawn at these points. These lines indicate asymptotes where the function is undefined. The function is cyclical with a period of π.

Transformations of Trigonometric Graphs

Trigonometric graphs can be transformed in four ways: translation, dilation, reflection, and rotation.
Dilation is the stretching or shrinking of the graph. When the function is multiplied by a constant factor, it changes the amplitude. For example, in y = A sin(θ), A is the amplitude.
Translation is the shifting of the function’s graph along the x or y axis. This is achieved by adding or subtracting a constant from the functions. For example, in y = sin(θ) + C, C translates the graph vertically.
Reflection is the flipping of the graph over the x or y axis. Multiplying the function by -1 will achieve a reflection. For example, y = -sin(θ) reflects the sin graph in the x-axis.
Rotation is achieved by adding or subtracting a constant from the angle θ. It shifts the phase of the function. For example, y = sin(θ + B) will shift the sin graph to the left if B > 0, and to the right if B < 0. B is referred to as the phase shift.

By understanding these rules and their effects on the graph of trigonometric functions, it becomes easier to interpret and sketch these graphs. Remember to always consider the effect of each transformation individually and then apply them in combination.