An exponential function is of the form: $y = a \cdot b^x$ Where:

• $\ y$ is the dependent variable.
• $\ x$ is the independent variable.
• $\ a$ is the initial value (the value of $\ y$ when $\ ( x = 0 ).$
• $\ b$ is the base of the exponential function.
• If $\ ( b > 1 )$, the function represents exponential growth.
• If $( 0 < b < 1$), the function represents exponential decay.

• Domain: The domain of an exponential function is all real numbers ($\ x \in \mathbb{R}$).
• Range:
• For $\ a > 0$ and $\ b > 0$, the range is $\ y > 0 .$
• For $\ a < 0$ and $\ b > 0$, the range is $\ y < 0 .$

• Intercepts:
• The $y$-intercept occurs at $\ x = 0$, giving $\ y = a \cdot b^0 = a .$
• Exponential functions typically do not have $x$-intercepts unless the function is horizontally shifted.

• Asymptote:
• The graph of an exponential function has a horizontal asymptote, usually at $\ y = 0$ (the $x$-axis).

• Growth and Decay:
• Exponential Growth: If $\ b > 1$, the function increases rapidly as $\ x$ increases.
• Exponential Decay: If $\ 0 < b < 1$, the function decreases rapidly as $\ x$ increases.

When a quantity grows at a rate proportional to its current value, it follows an exponential growth model: $y = y_0 \cdot e^{kt}$ Where:

• $\ y_0$ is the initial quantity.
• $\ k$ is the growth rate constant $( k > 0 )$ for growth).
• $\ t$ is time.

When a quantity decreases at a rate proportional to its current value, it follows an exponential decay model: $y = y_0 \cdot e^{-kt}$ Where:

• $\ y_0$ is the initial quantity.
• $\ k$ is the decay rate constant ($\ k > 0$ for decay).
• $\ t$ is time.

Example 1: Modelling Population Growth

• Problem: A population of bacteria doubles every $3$ hours. If there are initially $100$ bacteria, find the population after $9$ hours.
• Solution:
• The population follows an exponential growth model $\ P(t) = P_0 \cdot 2^{t/3}$, where $\ P_0 = 100$ and $\ t$ is the time in hours.
• After $9$ hours $t = 9$: $P(9) = 100 \cdot 2^{9/3} = 100 \cdot 2^3 = 100 \cdot 8 = 800$
• The population after $9$ hours is $800$ bacteria.

Example 2: Radioactive Decay

• Problem: A radioactive substance decays at a rate of $5$% per year. If the initial amount is $200$ grams, find the amount remaining after $10$ years.
• Solution:
• The decay model is $\ A(t) = A_0 \cdot e^{-kt} \, where \ k = 0.05$ and $\ A_0 = 200$ $grams$.
• After $10$ years ($t = 10$): $A(10) = 200 \cdot e^{-0.05 \cdot 10} = 200 \cdot e^{-0.5} \approx 200 \cdot 0.6065 = 121.3 \text{ grams}$
• The remaining amount after $10$ years is approximately $121.3$ grams.

• Exponential functions describe processes where growth or decay occurs at a rate proportional to the current value.
• The base $\ e$ (approximately 2.718) is particularly important in natural exponential functions, which have applications in many scientific fields.
• Understanding the transformation, graphing, and real-world applications of exponential functions is crucial for analysing growth and decay scenarios in mathematics, science, and finance.