Question: A linguist analyzes the frequency of a linguistic pattern with $ h(x) = x^2 - 2x + m $. If the frequency at $ x = 5 $ is 12, determine $ m $. - Veritas Home Health

Understanding the Context

In computational linguistics and language frequency analysis, mathematical models help uncover meaningful patterns in language behavior. One such model is defined by the quadratic function $ h(x) = x^2 - 2x + m $, where $ x $ represents a linguistic metric—such as word frequency rank, sentence length, or contextual usage intensity—and $ h(x) $ reflects the observed frequency or intensity.

Recently, a linguist investigated a recurring phrase pattern and found that at position $ x = 5 $, the observed frequency is exactly 12. To determine the unknown constant $ m $, meaningful algebraic analysis is essential.

The Problem: Find $ m $ Given $ h(5) = 12 $

Given the function
$$
h(x) = x^2 - 2x + m
$$
we substitute $ x = 5 $ and set $ h(5) = 12 $:

Key Insights

$$
h(5) = (5)^2 - 2(5) + m = 12
$$

Simplify:

$$
25 - 10 + m = 12
$$

$$
15 + m = 12
$$

Solving for $ m $:

Final Thoughts

$$
m = 12 - 15 = -3
$$

Why This Matters in Linguistics

Understanding constants like $ m $ is crucial in modeling linguistic behavior. This parameter may represent baseline frequency influence, contextual weight, or an adjustment factor tied to linguistic theory. Once $ m $ is determined, the model $ h(x) = x^2 - 2x - 3 $ provides precise predictions for pattern frequency across different linguistic contexts.

Final Answer

The value of $ m $ that ensures $ h(5) = 12 $ is $ oxed{-3} $.