A real-world function model translates a situation into mathematics. The input is usually a
measurable quantity, such as time, number of units, distance, or price. The output is the value
we want to predict or optimize, such as cost, profit, height, population, or revenue.
1. What is a function model?
A function model has the form
\[
y=f(x),
\]
where \(x\) is the input and \(y\) is the output. The model is useful only when the variables
have a clear meaning. For example, if \(x\) represents units sold and \(P(x)\) represents profit,
then \(P(100)\) is the predicted profit after selling \(100\) units.
2. Linear models
A linear function has the form
\[
f(x)=mx+b.
\]
The number \(m\) is the rate of change, and \(b\) is the starting value. Linear models are useful
for constant-speed motion, fixed-plus-variable costs, simple salary models, and direct trends.
For example, a distance model can be written as
\[
d(t)=d_0+vt,
\]
where \(d_0\) is the starting distance and \(v\) is speed.
3. Cost, revenue, and profit
In business applications, cost often has a fixed part and a variable part:
\[
C(x)=F+vx.
\]
Revenue is often modeled as
\[
R(x)=px,
\]
where \(p\) is the selling price per unit. Profit is revenue minus cost:
\[
P(x)=R(x)-C(x)=px-(F+vx).
\]
A break-even point occurs where profit is zero:
\[
P(x)=0.
\]
At break-even, revenue exactly equals cost.
4. Quadratic models and optimization
A quadratic model has the form
\[
f(x)=ax^2+bx+c.
\]
Quadratics are common in revenue models, projectile motion, area problems, and optimization
problems. The vertex gives the maximum or minimum value. The vertex input is
\[
x_v=-\frac{b}{2a}.
\]
If \(a<0\), the parabola opens downward and the vertex gives a maximum. If \(a>0\), the parabola
opens upward and the vertex gives a minimum.
5. Projectile motion
The height of a projectile can be modeled by
\[
h(t)=h_0+v_0t-\frac12gt^2.
\]
Here \(h_0\) is the initial height, \(v_0\) is the initial vertical velocity, and \(g\) is the
acceleration due to gravity. Since this is a quadratic model with negative leading coefficient,
the graph opens downward. The maximum height occurs at
\[
t_{\max}=\frac{v_0}{g}.
\]
The landing time is found by solving
\[
h(t)=0.
\]
6. Exponential population models
Population growth is often modeled exponentially:
\[
P(t)=P_0(1+r)^t.
\]
Here \(P_0\) is the initial population and \(r\) is the growth rate per time period written as a
decimal. For example, a \(5\%\) growth rate means \(r=0.05\). The doubling time is found by
solving
\[
P_0(1+r)^t=2P_0.
\]
Dividing by \(P_0\) gives
\[
(1+r)^t=2.
\]
Taking logarithms gives
\[
t=\frac{\ln 2}{\ln(1+r)}.
\]
7. Interpreting model answers
In modeling problems, the final answer must be interpreted. A value such as \(x=72\) means
different things in different contexts. It might mean \(72\) units sold, \(72\) seconds, \(72\)
years, or \(72\) kilometers. The mathematical answer is complete only when it is connected back
to the real situation.
8. Modeling assumptions
Every model makes assumptions. A linear distance model assumes constant speed. A simple profit
model assumes the selling price and variable cost stay constant. A basic exponential population
model assumes a constant growth rate. A projectile model ignores air resistance unless the model
is modified. These assumptions are not failures; they define when the model is reasonable.
9. Summary table
10. Final checklist
- Identify the input variable.
- Identify the output variable.
- Choose a model type: linear, quadratic, exponential, or another function.
- Substitute the known quantities into the model.
- Evaluate the function, solve for a target, or optimize the model.
- Check whether the answer is reasonable in context.
- Write the final interpretation using real-world units.
