Section 1.2: Mathematical Models: A Catalog of Essential Functions

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MATH161

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What is This Section About?

Have you ever wondered how scientists predict population growth, how engineers model the trajectory of a rocket, or how economists forecast market trends? They all use mathematical models—functions that describe real-world phenomena. This section is your introduction to the essential toolkit of functions that mathematicians and scientists use to model everything from temperature changes to radioactive decay.

Think of this section as building your “function vocabulary.” Just as a writer needs to know different literary forms (poetry, prose, drama), a mathematician needs to know different function families (linear, polynomial, exponential, trigonometric). Each function type has its own personality, its own strengths, and its own applications.

Why does this matter? Because the real world doesn’t come with instructions saying “use a quadratic function here.” You need to recognize patterns in data and phenomena, then select the right mathematical tool. This section teaches you not just what these functions are, but when and why to use each one.

The Big Picture: What You’ll Learn

By the end of this section, you’ll be able to:

1. Define what a mathematical model is and explain its role in describing real-world phenomena
2. Identify and construct linear models from data, understanding slope as rate of change
3. Recognize polynomial models (quadratic, cubic, etc.) and their characteristic shapes
4. Work with power and root functions, understanding how exponents affect behavior
5. Apply reciprocal and inverse square laws to physical phenomena
6. Understand rational and algebraic functions and their domain restrictions
7. Use trigonometric functions to model periodic phenomena
8. Distinguish between exponential and logarithmic functions for growth and decay

Think of this as assembling your mathematical toolbox—each function type is a specialized tool for specific jobs.

Core Concepts

What is a Mathematical Model?

A mathematical model is an idealized description of a real-world system using mathematical concepts and language. Models help us:

• Understand complex systems by simplifying them
• Predict future behavior based on current data
• Test hypotheses without expensive experiments

Key Insight: All models involve simplifying assumptions. A map is useful precisely because it’s not identical to the territory—it highlights what matters and ignores irrelevant details.

The modeling process:

1. Identify variables: What are you measuring? (independent vs. dependent)
2. Make assumptions: What can you safely ignore? (frictionless surfaces, uniform temperatures, etc.)
3. Choose a function type: Based on the pattern you observe
4. Fit parameters: Use data to determine specific coefficients
5. Test and refine: Does the model match reality? If not, iterate!

Linear Models: Constant Rate of Change

Form: $y = mx + b$

When to use it: When something changes at a constant rate

Key parameters:

• $m$ = slope = rate of change (how much $y$ changes per unit change in $x$)
• $b$ = $y$-intercept = value when $x = 0$

Real-world examples:

• Temperature vs. altitude (decreases by ~10°C per km)
• CO₂ concentration over time (increasing approximately linearly)
• Distance traveled at constant speed

In plain English: Linear models say “for every step I take in $x$, I always take the same size step in $y$.” The relationship is proportional and predictable.

Building a linear model from two points:

If you know $(x_1, y_1)$ and $(x_2, y_2)$:

1. Find slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$
2. Use point-slope form: $y - y_1 = m(x - x_1)$
3. Simplify to slope-intercept form if needed

Polynomial Models: Curves and Bends

General form: $P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$

The degree $n$ determines the function’s behavior:

Quadratic ($n=2$): $f(x) = ax^2 + bx + c$

• Graph: Parabola (U-shaped)
• Opens up if $a > 0$, opens down if $a < 0$
• Applications: Projectile motion, area optimization, profit models

Cubic ($n=3$): $f(x) = ax^3 + bx^2 + cx + d$

• Graph: S-shaped curve with possible local max and min
• Applications: Volume relationships, more complex motion

Key insight: Higher degree polynomials can model more complex behavior with more “turns” in the graph, but they require more data points to determine all coefficients.

Power Functions: The Foundation

Form: $f(x) = x^a$ where $a$ is any real number

Special cases:

• $a = 1$: Linear ($f(x) = x$)
• $a = 2$: Quadratic ($f(x) = x^2$)
• $a = 1/2$: Square root ($f(x) = \sqrt{x}$)
• $a = -1$: Reciprocal ($f(x) = \frac{1}{x}$)

Behavior depends on $a$:

• Even integer exponents: Symmetric about $y$-axis, always positive for real $x$
• Odd integer exponents: Symmetric about origin, can be positive or negative
• Fractional exponents: Root functions with domain restrictions
• Negative exponents: Reciprocal relationships with vertical asymptotes

Root functions: $f(x) = \sqrt[n]{x} = x^{1/n}$

• Even roots ($\sqrt{x}$, $\sqrt[4]{x}$) require $x \geq 0$
• Odd roots ($\sqrt[3]{x}$, $\sqrt[5]{x}$) are defined for all real $x$

Reciprocal and Inverse Square Functions

Reciprocal: $f(x) = \frac{1}{x}$

• Domain: All real numbers except $x = 0$
• Graph: Two branches (hyperbola), vertical asymptote at $x = 0$, horizontal asymptote at $y = 0$
• Applications: Boyle’s Law ($P = \frac{C}{V}$), average speed calculations

Inverse Square: $f(x) = \frac{1}{x^2}$

• Graph: Similar to reciprocal but always positive
• Applications: Gravitational force, light intensity, electromagnetic radiation
• Physical meaning: Many natural phenomena weaken with the square of distance

Key Insight: Inverse relationships model situations where increasing one quantity decreases another. The “inverse square” pattern appears throughout physics because it relates to spreading over the surface of a sphere (area = $4\pi r^2$).

Rational Functions: Ratios of Polynomials

Form: $f(x) = \frac{P(x)}{Q(x)}$ where $P$ and $Q$ are polynomials

Domain restriction: Exclude all values where $Q(x) = 0$

Key features:

• Vertical asymptotes: Where the denominator equals zero
• Horizontal asymptotes: Determined by comparing degrees of numerator and denominator
• Applications: Chemical concentration, optical lens formulas, economic equilibrium

Example: $f(x) = \frac{2x-1}{x^2-4}$

• Denominator factors as $(x-2)(x+2)$
• Domain: All real numbers except $x = 2$ and $x = -2$
• Vertical asymptotes at $x = 2$ and $x = -2$

Algebraic Functions: Built from Algebra

Algebraic functions are constructed from polynomials using the algebraic operations: addition, subtraction, multiplication, division, and taking roots.

Examples:

• $f(x) = \sqrt{x^2 - 4}$ (combines polynomial and root)
• $g(x) = \frac{x}{\sqrt{x+1}}$ (combines rational and root)

Why this matters: Many real-world relationships involve multiple operations. Algebraic functions provide the framework for modeling these complex relationships.

Trigonometric Functions: The Periodic Family

The six trigonometric functions: $\sin x$, $\cos x$, $\tan x$, $\csc x$, $\sec x$, $\cot x$

Key properties:

• Periodic: They repeat their values at regular intervals
• Period: $2\pi$ for sine and cosine, $\pi$ for tangent
• Range: $[-1, 1]$ for sine and cosine, all reals for tangent
• IMPORTANT: In calculus, always use radian measure, not degrees!

Applications:

• Ocean tides (periodic rise and fall)
• Sound waves (periodic pressure variations)
• Seasonal temperature variation
• AC electrical current

In plain English: Trigonometric functions model anything that cycles repeatedly. If your phenomenon “comes back” to where it started, trigonometry is likely involved.

Exponential and Logarithmic Functions: Growth and Decay

Exponential: $f(x) = b^x$ where $b > 0$, $b \neq 1$

• Domain: All real numbers
• Range: $(0, \infty)$ (always positive)
• Growth: If $b > 1$, function grows without bound
• Decay: If $0 < b < 1$, function approaches zero

Logarithmic: $g(x) = \log_b x$ (inverse of exponential)

• Domain: $(0, \infty)$ (only positive numbers)
• Range: All real numbers
• Relationship: $y = b^x \iff x = \log_b y$

Applications:

• Exponential: Population growth, compound interest, radioactive decay, epidemic spread
• Logarithmic: pH scale, Richter scale (earthquakes), decibel scale (sound)

Key Insight: Exponential functions grow (or decay) multiplicatively rather than additively. Each step multiplies by a constant factor rather than adding a constant amount. This makes them much more powerful than polynomial growth.

Worked Examples

Example 1: Building a Linear Model from Data

Problem: The temperature at ground level (altitude = 0 km) is 20°C, and at 1 km altitude, it’s 10°C. Build a linear model for temperature $T$ as a function of altitude $h$.

Thought Process: Temperature changes at a constant rate with altitude, so a linear model is appropriate. I have two data points, which is exactly what I need to determine slope and intercept.

Solution:

Step 1: Set up the linear model form \(T = mh + b\)

Step 2: Use the first data point (0, 20) When $h = 0$, $T = 20$: \(20 = m(0) + b\) \(b = 20\)

Step 3: Use the second data point (1, 10) When $h = 1$, $T = 10$: \(10 = m(1) + 20\) \(m = -10\)

Step 4: Write the complete model \(T = -10h + 20\)

Interpretation: The slope $m = -10$ means temperature decreases by 10°C for every kilometer of altitude gained. The intercept $b = 20$ is the ground-level temperature.

Key Takeaway: The slope in a linear model represents the rate of change—always interpret it in the context of the problem’s units!

Example 2: Quadratic Model for Projectile Motion

Problem: A ball is dropped from a height of 450 meters. Data suggests the height $h$ (in meters) follows a quadratic pattern with time $t$ (in seconds): $h = -4.90t^2 + 0.96t + 449.36$. When does the ball hit the ground?

Thought Process: The ball hits the ground when $h = 0$. I need to solve a quadratic equation. Since we’re modeling real-world time, only the positive solution makes sense.

Solution:

Step 1: Set height equal to zero \(-4.90t^2 + 0.96t + 449.36 = 0\)

Step 2: Apply the quadratic formula \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

where $a = -4.90$, $b = 0.96$, $c = 449.36$

Step 3: Calculate the discriminant \(b^2 - 4ac = (0.96)^2 - 4(-4.90)(449.36)\) \(= 0.9216 + 8807.456 = 8808.378\)

Step 4: Solve for $t$ \(t = \frac{-0.96 \pm \sqrt{8808.378}}{2(-4.90)}\) \(t = \frac{-0.96 \pm 93.85}{-9.80}\)

Taking the positive root: \(t = \frac{-0.96 + 93.85}{-9.80} \approx 9.48 \text{ seconds}\)

Interpretation: The ball hits the ground after approximately 9.48 seconds. The negative solution (-9.58) is physically meaningless in this context.

Key Takeaway: Quadratic models naturally arise from accelerating motion (gravity causes constant acceleration). Always check that your solutions make physical sense!

Example 3: Inverse Square Law

Problem: The intensity of light from a source is inversely proportional to the square of the distance from the source. If the intensity is 100 lumens at 2 meters, what is the intensity at 5 meters?

Thought Process: “Inversely proportional to the square of distance” translates to $I = \frac{k}{d^2}$ where $k$ is a constant. I’ll use the given data to find $k$, then use that to find the intensity at the new distance.

Solution:

Step 1: Write the inverse square model \(I = \frac{k}{d^2}\)

Step 2: Find $k$ using the given data At $d = 2$ meters, $I = 100$ lumens: \(100 = \frac{k}{2^2} = \frac{k}{4}\) \(k = 400\)

Step 3: Use the model to find intensity at 5 meters \(I = \frac{400}{5^2} = \frac{400}{25} = 16 \text{ lumens}\)

Interpretation: At 5 meters (2.5 times farther), the intensity drops to 16 lumens—only 16% of the original intensity. Doubling the distance quarters the intensity.

Key Takeaway: Inverse square laws show up throughout physics because they model how things spread out in three-dimensional space. The intensity drops dramatically with distance.

Example 4: Choosing the Right Model

Problem: You’re given data showing population over time. In 1990, the population was 5,000. In 2000, it was 7,500. In 2010, it was 11,250. What type of model seems appropriate?

Thought Process: Let me check if this could be linear or exponential by examining the growth pattern.

Solution:

Check for linear growth:

• From 1990 to 2000: increase of 2,500 over 10 years
• From 2000 to 2010: increase of 3,750 over 10 years
• The increases are different, so not linear!

Check for exponential growth:

• From 1990 to 2000: ratio is $7,500/5,000 = 1.5$
• From 2000 to 2010: ratio is $11,250/7,500 = 1.5$
• The ratios are the same! This suggests exponential growth.

Model: $P(t) = P_0 \cdot (1.5)^{t/10}$ where $t$ is years since 1990

Key Takeaway: Linear models have constant differences; exponential models have constant ratios. Check both patterns when given data!

Practice Problems

Try these problems to solidify your understanding:

1. Linear Model: CO₂ levels at Mauna Loa were 338.7 ppm in 1980 and 404.2 ppm in 2016. Create a linear model and use it to estimate the level in 2005.

2. Quadratic Model: A ball’s height in meters is given by $h(t) = -4.9t^2 + 20t + 2$ where $t$ is in seconds. Find when the ball reaches its maximum height and what that height is.

3. Power Functions: Graph $f(x) = x^3$ and $g(x) = x^5$. How do their shapes differ near $x = 0$ and for large $\vert x\vert $?

4. Rational Functions: Find the domain of $f(x) = \frac{1}{1 - 2\cos x}$ and explain which values must be excluded.

5. Trigonometric Model: A harbor’s water depth varies sinusoidally with high tide at 12 meters and low tide at 4 meters. Write a function modeling depth as a function of time.

6. Exponential vs. Linear: One city’s population grows by adding 1,000 people per year. Another grows by 5% per year. If both start at 50,000, which will be larger after 20 years?

Hints:

• Problem 1: Find slope using the two points, then use point-slope form
• Problem 2: The maximum occurs at the vertex; use $t = -\frac{b}{2a}$
• Problem 3: Higher odd powers are flatter near the origin and steeper away from it
• Problem 4: Find where $1 - 2\cos x = 0$ and exclude those values
• Problem 5: Use $D(t) = A\sin(Bt) + C$ where $A$ is amplitude, $C$ is midline
• Problem 6: Write formulas for both and compare at $t = 20$

Key Reminders

As you work with mathematical models, keep these principles in mind:

✓ Match the model to the pattern – Linear for constant rate, exponential for constant ratio, periodic for cycles

✓ Check your domain – Rational functions, root functions, and logarithms all have restrictions

✓ Interpret parameters – The slope, intercept, period, and amplitude all have real-world meanings

✓ Test your model – Does it make sense for extreme values? Does it match the data?

✓ Remember the inverse relationship – Exponential and logarithmic functions undo each other

✓ Use radians in calculus – Trigonometric derivatives only work with radian measure

Why This Matters

Mathematical modeling is the bridge between pure mathematics and the real world. When you:

• Predict climate change using atmospheric models
• Design bridges using stress and load models
• Forecast epidemics using exponential growth models
• Analyze financial markets using trend models
• Study ocean tides using trigonometric models

…you’re using the function families from this section! Every scientific field relies on mathematical modeling. Learning to recognize which model fits which situation is a superpower that will serve you throughout your career in any quantitative field.