Question

periodic functions periodic function flexibility which of the following options correctly writes an equation for a cosine function, based on the given graph? (1 point) $y = -\cos(x)$ $y = \cos(x) + 1$ $y = -\cos(x) + 1$ $y = -\cos(x - 1) + 1$

Explanation:

Step1: Recall the general form of cosine function transformation

The general form of a cosine function is \(y = A\cos(B(x - C))+D\), where \(A\) is the amplitude, \(D\) is the vertical shift.

Step2: Analyze the vertical shift

The standard \(y = \cos(x)\) has a range \([- 1,1]\). The given graph has a range \([0,2]\). The mid - line of the given graph is \(y = 1\), so there is a vertical shift \(D = 1\).

Step3: Analyze the reflection

The standard \(y=\cos(x)\) has a maximum at \(x = 0\). The given graph has a minimum at \(x = 0\) (after considering the vertical shift). So, there is a reflection about the \(x\) - axis, which means \(A=-1\).

Step4: Check for horizontal shift

There is no horizontal shift (\(C = 0\)) as the key points (minima/maxima) occur at the same \(x\) - values (after accounting for vertical shift and reflection) as the basic transformed cosine function without horizontal shift.

Substituting \(A=-1\), \(B = 1\), \(C = 0\), \(D = 1\) into \(y=A\cos(B(x - C))+D\), we get \(y=-\cos(x)+1\)

Answer:

\(y =-\cos(x)+1\) (the third option)