Introduction

Exponential functions are magical; they appear everywhere—from biology and economics to computer science. But what exactly is an exponential function? Understanding the exponential parent function is key to mastering these concepts in AP® Precalculus section 2.3 Exponential Functions. Let’s break it down to see how they work.

What is an Exponential Parent Function?

An exponential parent function forms the basis for all other exponential functions. It’s like the original, pure version that helps us understand how these functions behave. The general form of an exponential function is:

f(x) = ab^x

Here, a represents the starting value, while b is the base that determines the growth or decay rate.

Key Characteristics of Exponential Functions

Growth and Decay

Exponential growth occurs when b > 1. Imagine compound interest: it starts small, but with time, the increase becomes noticeable because it’s compounding. On the other hand, exponential decay occurs when 0 < b < 1. This is like the half-life in radioactive decay where substances decrease by a fixed percentage over time.

Example:

Consider the exponential function f(x) = 2 \cdot 3^x. Is it showing growth or decay?

• Identify the base: Here, b = 3.
• Analyzing the base: Since 3 > 1, it’s growth.
• Conclusion: This function represents exponential growth.

Practice Problem: Does f(x) = 5(0.4)^x show growth or decay?

• Solution: Here, b = 0.4. Since 0 < b < 1, it’s decay.

Domain and Range

The domain of exponential functions is all real numbers. However, the range depends on the function itself. Generally, if a > 0, the range is (0, \infty), since exponential functions do not touch the x-axis.

Example:

Determine the domain and range for exponential function: f(x) = 5^x.

• Domain: All real numbers
• Range: Since the base and coefficient are both positive, the range is (0, \infty).

Practice Problem: Find the domain and range of f(x) = -2 \cdot 3^x.

• Solution: Domain: All real numbers; Range: (-\infty, 0).

Asymptotes

The asymptote of exponential functions is a horizontal asymptote. An asymptote is a line that the graph approaches but never touches.

Example:

For f(x) = 2 \cdot (0.5)^x, the asymptote is…

• Explanation: As x goes to infinity, y approaches 0.
• Conclusion: The horizontal asymptote is y = 0.

Practice Problem: What is the horizontal asymptote of f(x) = -4^x + 3?

• Solution: The horizontal asymptote is y = 3.

Behavior of the Graph

Which graph represents an exponential function? The graph of exponential functions is either always increasing or decreasing. This characteristic depends on whether the function exhibits growth or decay.

Example:

Sketch the graph of f(x) = 2^x.

• Identification: Since b = 2, the function is increasing.
• We can make an exponential function table by plugging in values for x:
  • x=-2 \rightarrow f(x)=1/4
  • x=-1 \rightarrow f(x)=1/2
  • x=-0 \rightarrow f(x)=1
  • x= 1 \rightarrow f(x)=2
  • x=2 \rightarrow f(x)=4
• Graph: The graph rises from left to right, showing growth.

Practice Problem: Describe the behavior of f(x) = (0.3)^x.

• Solution: The function is decreasing because b = 0.3.

Properties of an Exponential Function

Exponential functions share some unique properties:

• Concavity: They’re always concave up.
• Lack of Extrema: No local minimum or maximum points.
• Transformations: Moving graphs horizontally or vertically won’t change their core behavior.

Example:

Examine f(x) = 3^x.

• Concavity: It’s concave up.
• Extrema: There’s none.
• Conclusion: Transformations would not alter these properties.

Practice Problem: Analyze f(x) = e^{-x} (where e is Euler’s number).

• Solution: It’s concave up and lacks extrema.

Transformations of Exponential Functions

Transformations include shifts or changes in position. Horizontal shifts move the graph left or right. Vertical shifts move it up or down.

Example:

For f(x) = 2^x + 3, the graph shifts…

• Vertical Shift: Up by 3 units (due to +3).
• Explanation: The graph will rise above the regular 2^x graph.

Practice Problem: What’s the transformation for f(x) = (0.5)^{x - 2}?

• Solution: A right shift of 2 units.

Summary of Key Terms

Term	Definition
Exponential Function	A function with repeated multiplication, f(x) = ab^x.
Growth	When b > 1, results in an increasing function.
Decay	When 0 < b < 1, leads to a decreasing pattern.
Domain	All real numbers for exponential functions.
Range	Usually (0, \infty) if a > 0.
Asymptote	A line the graph nears but never touches.

Conclusion

Understanding exponential functions is crucial for your math journey. It helps in analyzing various real-world phenomena that show rapid changes. Engage, practice, and explore more to harness these insights fully.

Sharpen Your Skills for AP® Precalculus

Are you preparing for the AP® Precalculus exam? We’ve got you covered! Try our review articles designed to help you confidently tackle real-world math problems. You’ll find everything you need to succeed, from quick tips to detailed strategies. Start exploring now!

Need help preparing for your AP® Precalculus exam?

Albert has hundreds of AP® Precalculus practice questions, free response, and an AP® Precalculus practice test to try out.

Sharpen Your Skills for AP® Precalculus

Are you preparing for the AP® Precalculus exam? We’ve got you covered! Try our review articles designed to help you confidently tackle real-world math problems. You’ll find everything you need to succeed, from quick tips to detailed strategies. Start exploring now!

Need help preparing for your AP® Precalculus exam?

Albert has hundreds of AP® Precalculus practice questions, free response, and an AP® Precalculus practice test to try out.