How To Graph Trigonometric Functions | Trigonometry

A final example, '2 sin(x - π/4) + 3', incorporates all transformations: amplitude, period, vertical shift, and phase shift. The instructor walks through finding each parameter, plotting the midline and amplitude boundaries, and then using the phase shift and period to accurately draw the complete trigonometric wave.

The example 'sin(x - π/2)' demonstrates how to graph a function with a phase shift. The starting point shifts to π/2, and the period is added to this new starting point to find the end of one cycle. The five key points are then plotted relative to this shifted interval.

Further examples, such as '2 cos x - 1' and '-3 sin(1/3 x) + 4', combine amplitude, period, and vertical shift. The process involves first plotting the new midline, then determining the upper and lower bounds based on amplitude, and finally sketching the wave according to the period.

The video introduces the phase shift, represented by 'c' in the generic formula, which causes the graph to shift horizontally. The phase shift is found by setting the argument of the function (bx + c) to zero and solving for x. The starting point of the wave is then adjusted by this phase shift.

The video begins by introducing the basic graphs of sine x and cosine x, explaining their sinusoidal nature and how a negative sign in front of the function flips the graph across the x-axis. It emphasizes graphing one period or cycle of the wave.

The instructor demonstrates how to graph two cycles of sine and cosine waves, breaking down each cycle into five key points (four intervals) for easier plotting. This method helps in accurately sketching the periodic nature of these functions.

The concept of amplitude 'a' in the generic formula 'a sin(bx + c) + d' is introduced. Amplitude is defined as the number in front of the sine or cosine function, representing the vertical stretch or compression of the graph. It's always a positive value (absolute value of 'a').

The video illustrates how to graph functions with different amplitudes, such as '2 sin x' and '-3 cos x'. It explains that the amplitude determines the maximum and minimum y-values and how a negative 'a' value flips the graph vertically. The domain and range are also discussed in relation to amplitude.

The period of a trigonometric function is introduced, calculated as '2π / b', where 'b' is the coefficient of x. Examples like 'sin 2x' and '2 sin(1/2 x)' demonstrate how 'b' affects the horizontal stretching or shrinking of the graph, completing a cycle in a shorter or longer interval.

An example of '4 cos(πx)' is used to show how both amplitude and period can be transformed simultaneously. The steps involve identifying 'a' and 'b', calculating the period, and then plotting the five key points within that period based on cosine's typical starting point.

The concept of a vertical shift 'd' is explained using 'sin x + 3'. This shifts the entire graph up or down along the y-axis, creating a new midline around which the wave oscillates. The amplitude still dictates the vertical variation from this new midline.