Mutation involves the change of the structure of DNA within a gene. During the replication of DNA, mutation may occur resulting in a non-exact replica and thus an alternate DNA sequence. The alternate DNA sequence / codon would result in the production of an alternate amino acid, thus creating an alternate polypeptide. Therefore, the mutation has resulted in an alternate form of the gene.
For the mutation to be passed onto future generations, the mutation must occur in the gametes. During sexual reproduction the new allele will then be passed onto offspring.
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For the "example" are we just supposed to know off our heads a specific example, e.g. Bluegas's "For example a single mutation in the haemoglobin molecule leads to the production of a different amino acid (valine instead of glutamic acid) and produces the genetic change."