The correct option is the first option (Image 1), which represents the graph of the function:
$f\left(x\right)=-2\sin \left(x\right)$

Step-by-Step Analysis:
1. Identify the parent function:
The given function is a transformation of the parent sine function, $y=\sin \left(x\right)$. The standard sine curve has a period of $2\pi$ and passes through the key points:
- $(0,0)$
- $(\frac{\pi }{2},1)$
- $(\pi ,0)$
- $(\frac{3\pi }{2},-1)$
- $(2\pi ,0)$

2. Determine the transformations:
For the function $f\left(x\right)=-2\sin \left(x\right)$:
- The coefficient 2 acts as a vertical stretch factor, meaning the amplitude of the wave is $|-2|=2$. The peaks will reach 2 and the troughs will reach -2.
- The negative sign (-) indicates a vertical reflection across the x-axis. Thus, instead of rising to a peak first, the graph will start by dipping into a trough.

3. Calculate the key points of the transformed function:
Applying these transformations to the standard key points of the sine function:
- At $x=0$: $f\left(0\right)=-2\sin \left(0\right)=0$ ⇒ Point is $(0,0)$.
- At $x=\frac{\pi }{2}$: $f\left(\frac{\pi }{2}\right)=-2\sin \left(\frac{\pi }{2}\right)=-2\left(1\right)=-2$ ⇒ Point is $(\frac{\pi }{2},-2)$.
- At $x=\pi$: $f\left(\pi \right)=-2\sin \left(\mathrm{��}\right)=0$ ⇒ Point is $(\pi ,0)$.
- At $x=\frac{3\pi }{2}$: $f\left(\frac{3\pi }{2}\right)=-2\sin \left(\frac{3\pi }{2}\right)=-2(-1)=2$ ⇒ Point is $(\frac{3\pi }{2},2)$.
- At $x=2\pi$: $f\left(2\pi \right)=-2\sin \left(2\pi \right)=0$ ⇒ Point is $(2\pi ,0)$.

4. Match with the options:
- Option 1 (Image 1): Starts at $(0,0)$, moves down to $-2$ at $\frac{\pi }{2}$, passes through zero at $\pi$, rises to a peak of $2$ at $\frac{3\pi }{2}$, and finishes its period at $(2\pi ,0)$. This perfectly matches our calculated points.
- Option 2 (Image 2): Represents a cosine wave with amplitude 2 starting at $(0,2)$, corresponding to $y=2\cos \left(x\right)$.
- Option 3 (Image 3): Represents a standard sine wave stretched vertically, starting at $(0,0)$ and going up first to $2$, corresponding to $y=2\sin \left(x\right)$.
- Option 4 (Image 4): Represents a reflected cosine wave starting at $(0,-2)$, corresponding to $y=-2\cos \left(x\right)$.

Therefore, the first option (Image 1) is the correct graph.