As I understand:
The purpose of a down payment is to minimize risk to the lender. For a traditional mortgage, lenders are willing to lend up to 80% of the value of the house.
The appraisal determines the value of the house for this purpose.
Therefore:
If the appraisal is lower than the sale price, the lender is willing to lend less, and my down payment is higher -- assuming I still wanted the house, and the seller didn't make an adjustment. If the appraisal was 5k lower, the minimum down payment would be 4k more.
And the reverse:
If the appraisal is higher than the sale price, the lender is willing to lend more, and my down payment is lower. If the appraisal was 5k higher, the minimum down payment is 4k less.
Is my understanding correct? Is this the purpose of a down payment, and its relationship to the appraisal?