Question

determine the period of the following graph.

Explanation:

Step1: Understand the definition of period

The period of a periodic function is the horizontal length over which the function repeats.

Step2: Identify two consecutive corresponding points

For a sinusoidal - like graph (assuming it's a periodic function), we can look at the distance between two consecutive peaks or troughs.
Let's consider the first peak (the left - most peak before the \(y\) - axis). Suppose the \(x\) - coordinate of the first peak is \(x_1=- 4\pi\), and the \(x\) - coordinate of the next peak is \(x_2 = 0\).

Step3: Calculate the period

The period \(T\) is given by the formula \(T=x_2 - x_1\). Substituting \(x_1=-4\pi\) and \(x_2 = 0\) into the formula, we get \(T=0-(-4\pi)=4\pi\). We can also check with other corresponding points. For example, if we consider the troughs: the \(x\) - coordinate of one trough is \(x=-2\pi\) and the \(x\) - coordinate of the next trough is \(x = 2\pi\). Then \(T=2\pi-(-2\pi)=4\pi\).

Answer:

$4\pi$