Let's analyze the graph step-by-step: ### **Step 1: Identify the General Form** This looks like a sine or cosine function. The general formula is: \[ y = a \cos(bx + c) + d \] or \[ y = a \sin(bx + c) + d \] Let's use cosine, as the graph starts at a maximum (which is typical for cosine). --- ### **Step 2: Amplitude (\(a\))** The graph oscillates between \(1\) and \(-1\), so: \[ \text{Amplitude},\; a = 1 \] --- ### **Step 3: Vertical Shift (\(d\))** The midline is at \(y = 0\), so: \[ d = 0 \] --- ### **Step 4: Period and Frequency (\(b\))** The period of cosine is \(2\pi / b\). From the graph: - One full cycle occurs from \(x = 0\) to \(x = 2\pi\). So, \[ \text{Period} = 2\pi \implies \frac{2\pi}{b} = 2\pi \implies b = 1 \] --- ### **Step 5: Phase Shift (\(c\))** At \(x = 0\), the graph is at its maximum, so there is **no phase shift**: \[ c = 0 \] --- ### **Step 6: Write the Final Formula** Putting it all together: \[ y = \cos(x) \] --- ## **Final Answer** \[ \boxed{y = \cos(x)} \]