In goats, the development of the beard is due to a recessive gene. The following cross involving true-breeding goats was made and carried to the \(\mathrm{F}_{2}\) generation: Offer an explanation for the inheritance and expression of this trait, diagramming the cross. Propose one or more crosses to test your hypothesis.

In this case, the development of the beard in goats is determined by a recessive gene. A recessive gene is one that is expressed only when an individual has two copies of the same allele, both carrying the trait. In a breeding process, if only one copy of the allele is present, the dominant allele will prevail, and the trait will not be expressed.

A Punnett square is a graphical representation of a cross between two individuals, showing the possible genotypes their offspring could inherit. In this case, let's use "B" for the dominant allele (no beard) and "b" for the recessive allele (beard). Since these are true-breeding goats, it means that they have homozygous genotypes, either "BB" or "bb". The initial cross (P generation) can be represented as: P generation: BB x bb To create the F1 generation, we will cross the P generation: F1 generation (genotype): Bb Considering that beard development requires the presence of two recessive alleles, all F1 generation goats will not have beards because they only have one recessive allele. Next, cross two F1 generation goats to create the F2 generation: F1 generation: Bb x Bb The Punnett square for this cross will show the following possible genotypes for the F2 generation: $\begin{bmatrix} BB & Bb \\ Bb & bb \end{bmatrix}$ F2 generation: 1 BB, 2 Bb, 1 bb (genotypic ratio = 1:2:1)

The F2 generation will exhibit two possible phenotypes: goats with beards (bb) and goats without beards (BB and Bb). Based on the genotypic ratio of the F2 generation, we will have a phenotypic ratio of 3:1, where 75% of goats will not have beards, and 25% will have beards. This phenotypic ratio confirms the inheritance and expression of the trait.

To test the hypothesis, we can carry out two different crosses: 1. Cross an F2 goat with a beard (bb) with a true-breeding goat with a beard (bb). This cross will produce offspring with a genotype of bb, resulting in a 100% probability that all offspring will have beards. This will support the recessive inheritance hypothesis. 2. Cross an F2 goat without a beard (Bb) with a true-breeding goat without a beard (BB). The resulting Punnett square will show the following genotypic outcomes: $\begin{bmatrix} BB & Bb \\ BB & Bb \end{bmatrix}$ The resulting offspring will have genotypes of 50% BB and 50% Bb, which also supports the hypothesis of the recessive inheritance of the trait, as none of the offspring will have beards.