### Teacher’s Summary

This insightful essay demonstrates how mathematical functions are deeply embedded in our daily lives. By applying concepts such as linear, quadratic, exponential, trigonometric, and logarithmic functions to real-world scenarios, the author effectively bridges the gap between abstract mathematics and practical applications. The essay highlights the universal nature of functions and underscores the value of interdisciplinary thinking, making complex ideas accessible and relevant. The author’s ability to connect mathematical principles to both personal and societal contexts is particularly commendable.

**Grade:** A

## Math Functions: The Hidden Mathematical Language of Our Daily Lives

As a history major with a minor in mathematics at Howard University, I’ve come to appreciate how mathematical concepts, particularly functions, permeate our everyday experiences. It’s fascinating to see how the abstract ideas we discuss in class manifest in the world around us. Let’s explore how functions shape our daily lives, from the mundane to the extraordinary.

### What is a Function?

Before we dive in, let’s revisit the basic definition of a function. In mathematical terms, a function is a relation between a set of inputs (the domain) and a set of permissible outputs (the range), with the property that each input corresponds to exactly one output. It’s like a machine that takes in one value and spits out another, following a specific rule.

### Linear Functions in Daily Life

Linear functions, those that create straight lines when graphed, are perhaps the most common in our day-to-day experiences. I noticed this recently when splitting a dinner bill with friends. The total cost per person (y) is a linear function of the number of people (x): y = T/x, where T is the total bill. As the number of people increases, the cost per person decreases proportionally.

Another example I encountered was during my part-time job at the campus bookstore. The store’s revenue (R) is a linear function of the number of textbooks sold (x): R = px + C, where p is the price per book and C is the fixed costs. This simple model helped me understand why my manager was always so focused on sales volume.

### Quadratic Functions: From Parabolas to Projectiles

Quadratic functions, those darlings of high school algebra with their U-shaped graphs, show up in surprising places. My physics professor demonstrated how the path of a thrown ball follows a quadratic function. The height (h) of the ball is a function of time (t): h = -16t² + v₀t + h₀, where v₀ is the initial velocity and h₀ is the initial height.

I even applied this concept when helping my little brother improve his basketball shots. We used a quadratic function to model the optimal arc for his free throws!

### Exponential Functions: Growth and Decay

Exponential functions, where the variable is in the exponent, describe phenomena that increase or decrease at a rapid rate. I’ve been fascinated by how these functions model everything from population growth to the spread of information on social media.

During a recent history project on the COVID-19 pandemic, I used exponential functions to understand the initial spread of the virus. The number of cases (N) over time (t) followed the function N = N₀e^(rt), where N₀ is the initial number of cases, r is the growth rate, and e is the mathematical constant.

### Trigonometric Functions: Cycles and Waves

Trigonometric functions like sine and cosine model repetitive phenomena. I noticed their application when studying sound waves in my physics elective. The amplitude (A) of a sound wave at any given time (t) can be described by the function A = Amax * sin(2πft), where Amax is the maximum amplitude and f is the frequency.

This concept helped me understand why my roommate’s snoring seems to follow a predictable pattern every night!

### Logarithmic Functions: The Inverse of Exponential Growth

Logarithmic functions, the inverse of exponential functions, are less obvious in daily life but no less important. They’re crucial in understanding phenomena like earthquake magnitudes on the Richter scale or the perception of loudness in decibels.

I recently used logarithmic functions to help explain compound interest to my study group. The time (t) it takes for an investment to double follows the logarithmic function: t = log(2) / log(1 + r), where r is the interest rate.

### Conclusion: The Universal Language of Functions

As I continue my journey through both history and mathematics, I’m constantly amazed by how functions serve as a universal language for describing the world around us. From splitting dinner bills to understanding global pandemics, functions provide a powerful tool for modeling and predicting real-world phenomena.

This exploration has reinforced my belief in the importance of interdisciplinary thinking. By understanding the mathematical underpinnings of everyday events, we gain a deeper appreciation for the interconnectedness of all knowledge. Whether you’re a historian, a scientist, or anything in between, the language of functions offers a unique lens through which to view and understand our complex world.

As I look towards my future studies and career, I’m excited to continue discovering new applications of functions in unexpected places. After all, isn’t life itself just one complex, beautiful function waiting to be understood?

**References:**

• Larson, Ron, and Bruce H. Edwards. *Calculus of a Single Variable*. 10th ed., Cengage Learning, 2014.

• Stewart, James. *Essential Calculus: Early Transcendentals*. 2nd ed., Cengage Learning, 2013.

• “Functions and Their Real-Life Applications.” *Math is Fun*, www.mathisfun.com/algebra/functions.html.