There are a couple of techniques for this.
Here’s the first technique…
Create a new static block (Static Content > Add New), give it an appropriate title such as “home page static header” for example, and then paste the URL of your header image into the Image input box in the Background metabox. Leave the background image options at their defaults (Left, Top, No Repeat).
Now in the article editor above, click the Text tab and add this content…
<div style="height:385px"></div>
The header area has a default height of 150px, and so what we are doing in the above HTML is adding a further 385px of height to the static content, thus giving it a total height of 150 + 385 = 535px, the same height as the demo slideshow.
Save the static block.
Now go to the layout manager (Appearance > Layouts) and in the Headers table click on the Home Page header to edit its settings. Set its Primary content option to reference “home page static header” (or whatever you named the static content we just created). Save the header.
Back in the layout manager, under the “Layout & Template Defaults” heading, make sure the Home Page context is set to the “Home Page” layout.
Your home page should now be displaying your image in the header in place of the slideshow.
I’ll describe the second technique in the post below.