91Ó°ÊÓ

Exponential functions have a specific pattern of change. They grow or decay at a constant percent rate over equal intervals of time.
An exponential function in the form related to our exercise can be expressed as:
\[ P(t) = P_0 \times (1 + r)^t \]
Here, \(P_0\) is the initial value (pollution level when \(t = 0\)), \(P(t)\) is the value after \(t\) years, and \(r\) represents the rate of growth or decay as a decimal.
For instance, in part a of the exercise, the pollution level triples each year. This means the rate of growth, \(r\), is 200% or 2 when transformed into its exponential form. So, we get:
\[ P(t) = 150 \times 3^t \]
Similarly, exponential decay occurs when the rate is negative. In part c, pollution decreases by 7% each year. Thus, the equation becomes:
\[ P(t) = 150 \times 0.93^t \]
This reflects the continued reduction, multiplying the previous year's pollution level by 0.93 (since 1 - 0.07 = 0.93).

Linear functions are simpler than exponential functions. They change by a constant amount over equal intervals of time.
A linear function in the context of our exercise can be written as:
\[ P(t) = P_0 + mt \]
Here, \(P_0\) is the initial pollution level, \(t\) is time in years, and \(m\) is the constant rate of change.
For example, in part b of the exercise, pollution decreases by twelve units each year. This can be expressed as:
\[ P(t) = 150 - 12t \]
This shows that for every additional year, the pollution level decreases consistently by 12 units.
Similarly, part d of the exercise tells us the rate of change is constant at 1. This means the pollution level changes linearly, expressed as:
\[ P(t) = 150 + t \]
In this case, since the rate of change is +1, the pollution is increasing by 1 unit each year.

The rate of change is a crucial concept in both linear and exponential functions. It indicates how quickly a quantity, such as pollution level, increases or decreases over time.
For linear functions: The rate of change is constant. It's the slope of the line when graphed. In our exercises:

• For a decrease of 12 units each year, the rate is -12, as in:
  \[ P(t) = 150 - 12t \]
• For a consistent increase of 1 unit each year, the rate is 1, shown as:
  \[ P(t) = 150 + t \]

The constant rate ensures that every year, the change is predictable and uniform.
For exponential functions: The rate of change varies depending on the current value. It grows or shrinks in proportion to its current value:

• For a tripling pollution level yearly, the rate follows the function:
  \[ P(t) = 150 \times 3^t \]
• For a 7% yearly decrease, it follows:
  \[ P(t) = 150 \times 0.93^t \]

Understanding the rate of change helps in predicting future values and making informed decisions in various scenarios involving growth and decay.