Bounce.js

class BounceOnMount extends React.Component {
  constructor(props: any) {
    super(props);
    this.state = {
      bounceValue: new Animated.Value(props.start),
    };
  }
  render(): ReactElement {
    return this.props.children(this.state.bounceValue);
  }
  componentDidMount() {
    Animated.spring(
      this.state.bounceValue,
      {
        toValue: this.props.end,
        friction: this.props.friction,
      }
    ).start();
  }
}

Playground.js

class Playground extends React.Component {
  render(): ReactElement {
    return (
      <BounceOnMount start={1.5} end={0.8} friction={0.8}>
        {bounceValue => (
          <Animated.Image
            source={{uri: 'http://i.imgur.com/XMKOH81.jpg'}}
            style={{
              flex: 1,
              transform: [
                {scale: bounceValue},
              ]
            }}
          />
        )}
      </Bounce>
    );
  }
}