Exponential Function, Its Properties and Graph
Grade 10 Algebra — A Complete Interactive Lesson
§1. Review of Key Properties of Powers with Real Exponents
Before studying the exponential function, let us recall the fundamental properties of powers. For $a > 0$, $b > 0$, and any real numbers $x_1$, $x_2$:
| # | Property | Formula | Name |
|---|---|---|---|
| 1 | Product of powers with the same base | $$a^{x_1} \cdot a^{x_2} = a^{x_1 + x_2}$$ | Product Rule |
| 2 | Quotient of powers with the same base | $$\frac{a^{x_1}}{a^{x_2}} = a^{x_1 - x_2}$$ | Quotient Rule |
| 3 | Power of a power | $$(a^{x_1})^{x_2} = a^{x_1 \cdot x_2}$$ | Power Rule |
| 4 | Power of a product | $$(a \cdot b)^{x} = a^{x} \cdot b^{x}$$ | Product of Bases |
| 5 | Power of a fraction | $$\left(\frac{a}{b}\right)^{x} = \frac{a^{x}}{b^{x}}$$ | Quotient of Bases |
⚡ Critical Ordering Properties
- (6) $a^{x} > 0$ for all $x \in \mathbb{R}$ — powers are always positive
- (7) $a^{x} > 1$ if $a > 1$ and $x > 0$
- (8) If $a > 1$ and $x_1 < x_2$, then $a^{x_1} < a^{x_2}$ increasing
- (9) If $0 < a < 1$ and $x_1 < x_2$, then $a^{x_1} > a^{x_2}$ decreasing
§2. Definition of the Exponential Function
📌 Definition
A function of the form $y = a^x$, where $a$ is a fixed positive number and $a \neq 1$, is called an exponential function with base $a$.
Key Points:
- The base $a$ is a fixed positive number, $a \neq 1$
- The variable $x$ appears in the exponent
- The argument of the exponential function is the exponent
- The base is a fixed number, not a variable
Examples of Exponential Functions:
- $y = 2^x$ — base $a = 2 > 1$
- $y = 10^x$ — base $a = 10 > 1$
- $y = \left(\tfrac{1}{2}\right)^x$ — base $a = \tfrac{1}{2}$, where $0 < a < 1$
- $y = 0.1^x$ — base $a = 0.1$, where $0 < a < 1$
⚠️ Why is $a = 1$ excluded? If $a = 1$, then $y = 1^x = 1$ for all $x$. This is just the constant function $y = 1$, which is not interesting as an exponential function.