Exponential Functions | Definition, Properties & Graphs

Learning Objectives

By the end of this article, you should be able to:

• Define an exponential function of the form \( f(x) = a^x \) (where \( a > 0, a \neq 1 \)).
• Identify the domain, range, asymptote, and monotonicity of exponential functions.
• Sketch exponential graphs and interpret key features.
• Apply the laws of exponents to simplify exponential expressions.
• Apply graph transformations (translations, stretches, reflections) to exponential functions.
• Solve DSE-style problems involving exponential functions and real-world growth/decay models.

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1. Introduction:

Exponential functions appear every year in DSE Paper 1 Section A(2) and Section B. Questions may ask you to:

• Sketch or interpret the graph of an exponential function
• Apply transformations to exponential graphs
• Simplify expressions using laws of exponents
• Solve exponential equations or model growth/decay

These are essential skills that also appear in logarithmic functions.

2. Definition and General Form

An exponential function is a function of the form:

The base \( a \) is a positive constant, and the exponent \( x \) is the variable.

Key Properties

• Domain: All real numbers (\( \mathbb{R} \))
• Range: \( y > 0 \) (all positive real numbers)
• Passes through: \( (0, 1) \) because \( a^0 = 1 \)
• Asymptote: The x-axis (\( y = 0 \))
• Monotonicity:
  • If \( a > 1 \), the function is increasing (exponential growth).
  • If \( 0 < a < 1 \), the function is decreasing (exponential decay).

3. Laws of Exponents

To work with exponential functions, you must be comfortable with the laws of exponents:

• \( a^m \times a^n = a^{m+n} \)
• \( a^m \div a^n = a^{m-n} \)
• \( (a^m)^n = a^{mn} \)
• \( a^{-n} = \frac{1}{a^n} \)
• \( a^0 = 1 \)
• \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \)

4. Graphing Exponential Functions

To sketch the graph of \( y = a^x \):

• Identify the asymptote (\( y = 0 \)).
• Plot the intercept \( (0, 1) \).
• Plot a few other points, e.g., \( (1, a) \), \( (-1, \frac{1}{a}) \).
• Draw a smooth curve approaching the asymptote.

5. Transformations of Exponential Graphs

Transformations of \( y = a^x \) follow the same rules as other functions:

Transformation	Effect
\( y = a^x + k \)	Vertical shift by \( k \) (asymptote becomes \( y = k \))
\( y = a^{x-h} \)	Horizontal shift by \( h \)
\( y = k a^x \)	Vertical stretch/compression by factor \( k \)
\( y = a^{-x} \)	Reflection in y-axis (decay if \( a > 1 \))
\( y = -a^x \)	Reflection in x-axis

6. Exponential Growth and Decay Models

Exponential functions model real-world situations where quantities grow or decay at a constant percentage rate.

• Growth model: \( y = A(1 + r)^t \) where \( r > 0 \)
• Decay model: \( y = A(1 - r)^t \) where \( 0 < r < 1 \)

Here \( A \) is the initial amount, \( r \) is the rate per time period, and \( t \) is time.

7. Worked Examples

Example 1: Sketching an Exponential Graph

Question: Sketch \( y = 3^x \) and state its domain, range, and asymptote.

Example 2: Transformations

Question: Describe the transformation from \( y = 2^x \) to \( y = 2^{x-3} + 1 \).

Example 3: Exponential Equation

Question: Solve \( 2^{2x+1} = 8 \).

8. DSE-Style Practice Questions

9. Solutions with Explanations

9. Exercise

Click the following link to have
 An Exercise on Exponential Functions | Definition, Properties & Graphs

Key Takeaways

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Summary Checklist for Revision

• Definition: \( f(x) = a^x \)
• Domain and range
• Asymptote: \( y = 0 \)
• Increasing vs decreasing
• Laws of exponents
• Transformations: shifts, stretches, reflections
• Exponential equations: same base method
• Growth/decay models