Algebra 2

Modeling with Functions

Math isn’t just abstract — it’s a tool for understanding the real world. When you write a function to describe a real situation, that’s called mathematical modeling. Let’s see how different types of functions capture different real-world behaviors.

Part 1: Revenue, Cost, and Profit

Imagine you’re selling t-shirts. You need to decide on a price.

Revenue = price * quantity sold
Cost = fixed costs + variable costs
Profit = Revenue - Cost

Here’s the catch: if you raise the price, fewer people buy. Let’s say demand drops linearly: quantity = 100 - 2 * price.

Price ($)20

545

Fixed costs ($)200

100800

\text0 = p \times (100 - 2p) = -20^2 \cdot 2 + 100 \cdot 20

Try This

Find the sweet spot: Revenue (green) is a downward parabola. Cost (red) is a line. Profit (blue) is the gap between them. Drag the price slider and watch the vertical distance between Revenue and Cost change. The maximum profit occurs around price = $25-30. Can you see why setting the price too high OR too low hurts profit?

Part 2: Linear Growth — Steady and Predictable

Some real-world quantities grow at a constant rate. A car driving at constant speed, a worker earning hourly wages, water filling a pool at a steady rate:

y = \text0 \times x + \text0

Rate ($/hour)8

020

Starting amount ($)50

0100

y = 8x + 50

Connection

Real-world examples of linear models:

Taxi fare: $3 base + $2.50/mile
Cell phone plan: $30/month + $0.10/text
Filling a pool: starts with 200 gallons, adds 15 gallons/minute

Part 3: Exponential Growth — Starting Slow, Then Exploding

Population growth, compound interest, and viral spread all follow exponential patterns:

y = a \cdot b^x

Initial amount (a)5

120

Growth factor (b)1.2

1.052

y = 5 \cdot 1.2^x

Try This

Watch the crossover: At first, exponential growth is slower than linear growth (the gray line is above the red curve). But eventually the exponential curve rockets past. Try setting growth factor b = 1.5 and see how quickly it overtakes the linear model.

Part 4: Quadratic Models — Rise and Fall

Some situations have a natural peak: a ball thrown in the air, profit maximization, or the area of a fenced region with limited material:

y = -a(x - h)^2 + k

Curvature (a)0.5

0.13

Peak time (h)8

015

Peak value (k)50

10100

y = -0.5(x - 8)^2 + 50

Connection

Real-world quadratic models:

Projectile motion: A ball reaches maximum height then falls
Business profit: Revenue peaks at an optimal price, then declines
Fencing problem: Maximum area enclosed with a fixed perimeter

Part 5: Choosing the Right Model

How do you know which function fits your situation?

Pattern	Model	Function
Constant rate of change	Linear	y = mx + b
Constant percent change	Exponential	y = a * b^x
Rises then falls (or vice versa)	Quadratic	y = ax^2 + bx + c
Approaches a limit	Logarithmic	y = a * ln(x) + b

All four models start at similar values but behave very differently as x grows. The right model depends on the behavior of your data.

Try This

Ask yourself these questions:

Is the change constant? (Linear)
Does it keep getting faster and faster? (Exponential)
Does it peak and then decline? (Quadratic)
Does it grow quickly at first then level off? (Logarithmic)

Part 6: A Complete Example — Lemonade Stand

You run a lemonade stand. Your research shows:

Demand: You sell (80 - 10p) cups when the price is p dollars
Cost per cup: $0.50 for ingredients
Fixed costs: $20 for the stand rental

Price per cup ($)3

\text{Cups sold} = 80 - 10 \times 3

\text0 = 3 \times (80 - 10 \times 3)

\text0 = 20 + 0.5 \times (80 - 10 \times 3)

Challenge

Challenge: Use the graph and slider to answer:

At what price do you maximize revenue? (Hint: peak of the green curve)
At what price do you maximize profit? (Hint: peak of the blue curve — it’s NOT the same as maximum revenue!)
At what price do you break even (profit = 0)?
Why is the maximum profit price different from the maximum revenue price?

Wrapping Up

Concept	Key Takeaway
Mathematical model	A function that describes a real situation
Revenue	Price times quantity sold
Profit	Revenue minus cost
Choosing a model	Match the function to the data’s behavior
Parameters	Sliders represent real decisions (price, rate, etc.)

Mathematical modeling is how engineers design bridges, economists forecast markets, biologists track populations, and businesses set prices. The functions you’ve learned in algebra aren’t just abstract — they’re tools for understanding and predicting the real world.

Take the Quiz