Let's break down the properties of the given trigonometric function step by step.
1) **Amplitude**: It can be observed from the trigonometric function that the amplitude is 4. The amplitude of a trigonometric function is the absolute value of the coefficient of the sine or cosine function. This represents the maximum vertical distance the function reaches from its midline. In this case it is |-4| = 4.
2) **Period Length**: The period length of the function can be calculated by dividing 2π by the coefficient of x in the sin function. So, here it would be 2π / 5. The period length shows how long it takes for the function to complete one cycle.
3) **Distance between critical points**: The critical points are the maximum, minimum, and zeros of a trigonometric function. These occur for every π/2 radians for a sine or cosine function. Therefore, with our period of 2π / 5, the distance between these points is half of this period, which means it would be π / 5.
4) **Phase Shift**: The phase shift refers to the horizontal shift of the graph. It is given by the coefficient of pi divided by the coefficient of x in the sin function. From this function, the phase shift is (3π) / 5 = 3 / 5 to the right. A positive shift means the graph moved to the right.
5) **Vertical Shift**: The vertical shift of a trigonometric function is the constant term. Here, the vertical shift is -2. A negative vertical shift implies the graph is shifted downwards.
6) **Equation of the Midline**: The midline of a sinusoidal function is a horizontal line at the level of the vertical shift. In simple words, that's where the function centers on. In this function, the equation for the midline is y = -2.