The Hardy-Weinberg principle is a fundamental concept in population genetics that describes how allele frequencies in a population remain constant from generation to generation in the absence of evolutionary influences. This principle is represented mathematically by the equations:

\[p^2 + 2pq + q^2 = 1\]

where:

• p = frequency of the dominant allele
• q = frequency of the recessive allele
• p² = frequency of homozygous dominant individuals
• 2pq = frequency of heterozygous individuals
• q² = frequency of homozygous recessive individuals

In a given population, if the allele frequencies are known, the genotype frequencies can be calculated using these equations. For example, if the frequency of allele p is 0.2 and allele q is 0.8, the calculations would be as follows:

\[p^2 = (0.2)^2 = 0.04\]

\[2pq = 2 \times 0.2 \times 0.8 = 0.32\]

\[q^2 = (0.8)^2 = 0.64\]

These calculations can be repeated for different populations to determine their respective genotype frequencies. For instance, if allele frequencies for another population are 0.4 and 0.6, the calculations yield:

\[p^2 = (0.4)^2 = 0.16\]

\[2pq = 2 \times 0.4 \times 0.6 = 0.48\]

\[q^2 = (0.6)^2 = 0.36\]

Once the genotype frequencies are calculated, they can be graphically represented. The y-axis typically represents genotype frequencies, while the x-axis represents allele frequencies. As allele frequencies change, the corresponding genotype frequencies can be plotted, revealing trends in the population's genetic structure.

In populations at Hardy-Weinberg equilibrium, the graph of genotype frequencies will exhibit specific patterns. For instance, as the frequency of allele p increases, the frequency of homozygous dominant individuals (p²) also increases, while the frequency of heterozygous individuals (2pq) reaches a peak and then declines. Conversely, the frequency of homozygous recessive individuals (q²) decreases as p increases.

These trends illustrate that when allele frequencies are low, homozygous individuals are rare, while higher frequencies of alleles lead to a greater presence of homozygous individuals. Additionally, in a stable population, heterozygotes are expected to be more common than either homozygous genotype, reinforcing the significance of genetic diversity within populations.

Understanding these dynamics is crucial for predicting how populations will respond to environmental changes and for conservation efforts aimed at maintaining genetic diversity.