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:
A company's profit ($P$) increases by 500 dollars for each unit ($x$) sold, with fixed costs of 2000 dollars. This can be modeled as:
$$P(x)=500x−2000$$
Here, $m=500$ (profit per unit) and $c=−2000$ (fixed costs).
When identifying linear relationships, look for constant differences between consecutive y-values for equal x-value intervals.
Quadratic models represent relationships where the rate of change varies linearly with the independent variable. The general form is:
$f(x) = ax^2 + bx + c$ (where $a \neq 0$)
Key features:
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).
Students often forget that $a$ must not be zero in a quadratic function. If $a = 0$, the function becomes linear.
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 $0<a<1$): $$f(x) = ka^x + c$$
Natural exponential: $$f(x) = ke^{rx} + c$$
Where:
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.
The natural exponential form $ke^{rx}$ is often preferred in continuous growth/decay scenarios, particularly in physics and finance.
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