If you wanted to be certain that a trait would appear in the offspring of the plants or animals that you were breeding, what would you have to find out about the parents?

What genes each parent had for that trait. 

Example: In peas, the trait for green peas is dominant (G) and the trait for yellow peas (g) is recessive. If you want the offspring to definitely be yellow, then both parents have to be yellow, with the allele frequency of gg. If both parents were carriers of the yellow gene, but were green (Gg), then there is a 25% chance of having yellow offspring, the rest being green. If one parent is a carrier (Gg) and the other is yellow (gg), then there is a 50% chance of having either yellow or green offspring. If one parent is homozygous (two alleles of the same gene) dominant, then no matter who that parent is paired with, then the offspring will definitely be green. 

This can all be figured out through punnett squares

Amutation is basically the change in the gene, caused by alteration of single based units in the dna, or the genes or chromosomes were rearranged in the dna