Question

the following is a cosine graph that has been horizontally shifted.
find the equations for this graph with respect to the closest left shift and the closest right shift.
round your answers to two decimal places.
the equation for the closest left shift is:
$$ y = \cos ( x - \square ) $$
the equation for the closest right shift is:
$$ y = \cos ( x - \square ) $$

Explanation:

Step1: Recall the general form of a cosine function

The general form of a cosine function is \(y = A\cos(B(x - C))+D\). For the standard \(y=\cos(x)\), the maximum occurs at \(x = 0\), and the minimum occurs at \(x=\pi\).

Step2: Analyze the left - shift

For a left - shift, we want to find the value \(C\) such that the graph of \(y = \cos(x - C)\) is shifted. The standard \(y=\cos(x)\) has a minimum at \(x=\pi\). In the given graph, a minimum occurs at \(x=\frac{\pi}{2}\). Using the formula for the phase shift \(x - C=\pi\) (for the minimum of \(y = \cos(x - C)\)). Substituting \(x=\frac{\pi}{2}\), we get \(\frac{\pi}{2}-C=\pi\), so \(C=-\frac{\pi}{2}\approx - 1.57\)

Step3: Analyze the right - shift

The standard \(y = \cos(x)\) has a minimum at \(x=\pi\). In the given graph, another minimum (considering the right - shift) can be thought of in terms of the period. The period of \(y=\cos(x)\) is \(2\pi\). If we consider the "next" minimum in the positive \(x\) direction (relative to the standard position) for the shifted function \(y=\cos(x - C)\). The period of \(y=\cos(x)\) is \(T = 2\pi\). If we use the fact that for the function \(y=\cos(x - C)\), if we consider the minimum. Let's assume we can also use the fact that if we rewrite the shift. The standard \(y=\cos(x)\) and the shifted function. If we consider the general form \(y=\cos(x - C)\), and we know that the pattern repeats every \(2\pi\). If we consider the right - shift, we can also note that if we use \(x - C=-\pi\) (because \(\cos(-\pi)=- 1\)). If \(x=\frac{3\pi}{2}\), then \(\frac{3\pi}{2}-C=-\pi\), so \(C=\frac{3\pi}{2}\approx4.71\)

Answer:

The equation for the closest left shift is \(y = \cos(x+1.57)\) (since \(x-(-1.57)=x + 1.57\)) and the equation for the closest right shift is \(y=\cos(x - 4.71)\)