Let $ V $ denote a vector space with finite positive dimension, and let $ (A, A^*) $ denote a Leonard pair on $ V $. As is known, the linear transformations $ A $, $ A^* $ satisfy the Askey-Wilson relations $ A^2A^* − \betaAA^*A + A^*A^2 − gamma(AA^* + A^*A) − sigmaA^* = gamma^*A^2 + omegaA + etaI, A^2A − \betaA^*AA^* + AA^*2 − gamma^*(A^*A + AA^*) − sigma^*A = gammaA^*2 + omegaA^* + eta^*I $, for some scalars $ \beta $, $ gamma $, $ gamma^* $, $ sigma $, $ sigma^* $, $ omega $, $ eta $, $ eta^* $. The scalar sequence is unique if the dimension of $ V $ is at least 4. If $ c $, $ c^* $, $ t $, $ t^* $ are scalars and $ t $, $ t^* $ are not zero, then $ (tA + c, t^*A^* +c^*) $ is a Leonard pair on $ V $ as well. These affine transformations can be used to bring the Leonard pair or its Askey-Wilson relations into a convenient form. This paper presents convenient normalizations of Leonard pairs by the affine transformations, and exhibits explicit Askey-Wilson relations satisfied by them. Kyushu University 21st Century COE Program Development of Dynamic Mathematics with High Functionality 九州大学21世紀COEプログラム「機能数理学の構築と展開」