When editing your Activity or News Update, you'll see a Visibility module. Your first option in the Visibility module is to feature your Activity or News Update on your homepage in the large slider/carousel. The slider is the area at the top of your homepage where the title of an Activity or News Update is paired with a large image and small snippet of text. It is important to keep in mind that only five News Updates or Activities can be included in this slider at any one time. To feature your Activity or News Update just check the "Feature on Homepage" option.
After selecting to feature on the homepage, you MUST select a featured image, which you can find instructions on here.
Similarly, if you no longer wish for your Activity or News Update to be featured in the homepage slider, just uncheck the box and click the "Update" button and it will no longer appear in the slider.
This video will show you how to set a Featured Image: