Question

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 a cosine function

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

Step2: Analyze the amplitude and vertical shift

For \(y = \cos(x)\), when \(x = 0\), \(y=\cos(0)=1\). In the given graph, when \(x = 0\), \(y = 0\).
If we consider \(y=-\cos(x)+1\):
When \(x = 0\), \(y=-\cos(0)+1=- 1 + 1=0\)
The amplitude of \(y = \cos(x)\) is \(|A| = 1\). For \(y=-\cos(x)+1\), the reflection (due to the negative sign in front of \(\cos(x)\)) and vertical shift (by \(+1\)) are considered.
The standard \(y = \cos(x)\) has a range \([-1,1]\). For \(y=-\cos(x)+1\), the range is \([0,2]\) which matches the graph (since the minimum value is \(0\) and maximum is \(1 + 1=2\) in terms of the vertical scale shown)

Answer:

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