Linear Models
Linear models are fundamental in mathematical modeling, representing relationships where one variable changes at a constant rate with respect to another. The general form of a linear function is:
$f(x) = mx + c$
Where:
- $m$ is the slope (rate of change)
- $c$ is the y-intercept (initial value)
A company's profit ($P$) increases by 500 for each unit ($x$) sold, with fixed costs of 2000. This can be modeled as:
$$P(x)=500x−2000$$
Here, $m=500$ (profit per unit) and $c=−2000$ (fixed costs).
TipWhen identifying linear relationships, look for constant differences between consecutive y-values for equal x-value intervals.
Quadratic Models
Quadratic models represent relationships where one variable changes at a rate that is proportional to another variable. The general form is:
$f(x) = ax^2 + bx + c$ (where $a \neq 0$)
Key features:
- Parabolic shape
- Vertex (maximum or minimum point)
- Axis of symmetry
- y-intercept at $(0, c)$
The height ($h$) of a ball thrown upwards after $t$ seconds can be modeled by:
$h(t) = -4.9t^2 + 20t + 1.5$
Here, $a = -4.9$ (due to gravity), $b = 20$ (initial velocity), and $c = 1.5$ (initial height).
Common MistakeStudents often forget that $a$ must not be zero in a quadratic function. If $a = 0$, the function becomes linear.
Exponential Growth and Decay Models
Exponential models represent situations where a quantity grows or decays by a constant percentage over equal intervals. The general forms are:
Growth: $$f(x) = ka^x + c$$
Decay (for $a > 0$) : $$f(x) = ka^{-x} + c$$
Natural exponential: $$f(x) = ke^{rx} + c$$
Where:
- $k$ is the initial value
- $a$ is the growth/decay factor
- $r$ is the growth/decay rate
- $c$ is the horizontal asymptote
A population of bacteria doubles every 3 hours. Starting with 1000 bacteria, the population ($P$) after $t$ hours can be modeled as:
$P(t) = 1000 \cdot 2^{t/3}$
Here, $k = 1000$, $a = 2$, and the exponent is $t/3$ because doubling occurs every 3 hours.
NoteThe natural exponential form $ke^{rx}$ is often preferred in continuous growth/decay scenarios, particularly in physics and finance.
Direct and Inverse Variation
These models represent relationships where one variable is directly or inversely proportional to a power of another variable. The general form is:
$$f(x) = ax^n$$
where $n \in \mathbb{Z}$
- Direct variation: $n > 0$
- Inverse variation: $n< 0$
The area $(A)$ of a circle is directly proportional to the square of its radius ($r$): $A=\pi r^2$ (Here, $a=\pi$ and $n=2$ )
Boyle's Law states that the pressure $(P)$ of a gas is inversely proportional to its volume ( $V$ ): $P=\frac{k}{V}$ or $P=k V^{-1}$ (Here, $a=k$ and $n=-1$ )
When graphing, direct variation with odd $n$ passes through the origin, while even $n$ creates a U-shaped curve. Inverse variation always creates a hyperbola.
Cubic Models
Cubic models are useful for representing more complex relationships, often involving inflection points. The general form is:
$$f(x) = ax^3 + bx^2 + cx + d$$
Key features:
- S-shaped curve (for $a > 0$)
- Possible multiple roots
- One or two turning points
