8.1 Exponential Functions and Their Properties

Exponential functions are like supercharged growth machines. They can model everything from compound interest to population explosions, showing how things can change rapidly over time. Understanding their properties is key to grasping their power and versatility.

In this part of the chapter, we'll dive into the nuts and bolts of exponential functions. We'll explore their definition, behavior, and how to work with them. Get ready to see how these functions can describe real-world phenomena in mind-blowing ways.

Exponential functions

Definition and components

Top images from around the web for Definition and components

• 

  Exponential Functions | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Equations of Exponential Functions | College Algebra Corequisite View original

  Is this image relevant?

• 

  Graphs of Exponential Functions | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Exponential Functions | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Equations of Exponential Functions | College Algebra Corequisite View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and components

• 

  Exponential Functions | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Equations of Exponential Functions | College Algebra Corequisite View original

  Is this image relevant?

• 

  Graphs of Exponential Functions | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Exponential Functions | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Equations of Exponential Functions | College Algebra Corequisite View original

  Is this image relevant?

1 of 3

• An exponential function is a function of the form $f(x) = a \cdot b^x$ where $a \neq 0$, $b > 0$, and $b \neq 1$
• In the exponential function $f(x) = a \cdot b^x$, $a$ is the initial value or y-intercept, $b$ is the base, and $x$ is the exponent or power
• The domain of an exponential function is all real numbers, and the range is all positive real numbers

Behavior and properties

• If the base $b$ is greater than 1, the function increases exponentially (exponential growth)
• If the base $b$ is between 0 and 1, the function decreases exponentially (exponential decay)
• The y-intercept of an exponential function is always $(0, a)$, where $a$ is the initial value
• Exponential functions exhibit a property called "exponential growth" or "exponential decay" depending on the value of the base $b$

Evaluating exponential expressions

Evaluating exponential functions

• To evaluate an exponential function, substitute the given value of $x$ into the function and simplify using the order of operations (PEMDAS)
  • Example: Given $f(x) = 2 \cdot 3^x$, find $f(2)$
    • $f(2) = 2 \cdot 3^2 = 2 \cdot 9 = 18$

Simplifying expressions with exponents

• When simplifying expressions with exponents, apply the properties of exponents:
  • Product rule: $a^m \cdot a^n = a^{m+n}$
  • Quotient rule: $a^m \div a^n = a^{m-n}$ $(a \neq 0)$
  • Power rule: $(a^m)^n = a^{mn}$
  • Zero exponent rule: $a^0 = 1$ $(a \neq 0)$
  • Negative exponent rule: $a^{-n} = \frac{1}{a^n}$ $(a \neq 0)$
• Simplify expressions involving rational exponents using the nth root property: $\sqrt[n]{a} = a^{\frac{1}{n}}$
  • Example: $\sqrt[3]{8} = 8^{\frac{1}{3}} = 2$
• Evaluate exponential expressions containing variables using substitution and the properties of exponents
  • Example: Given $x = 2$ and $y = 3$, evaluate $2x^2y^3$
    • $2x^2y^3 = 2(2)^2(3)^3 = 2 \cdot 4 \cdot 27 = 216$

Graphing exponential functions

Creating graphs

• To graph an exponential function, create a table of values by substituting $x$-values into the function and calculating the corresponding $y$-values
• Plot the points from the table of values on a coordinate plane and connect them with a smooth curve

Analyzing graphs

• Exponential functions are not linear; they curve upward (exponential growth) or downward (exponential decay) depending on the base value
• The graph of an exponential function will never touch the $x$-axis because the range is always positive
• Analyze the graph to identify key features such as the $y$-intercept, asymptotes, and end behavior
• Compare the graphs of exponential functions with different base values to understand how the base affects the growth or decay rate
  • Example: Graph $f(x) = 2^x$ and $g(x) = (\frac{1}{2})^x$ on the same coordinate plane to compare their behavior

Properties of exponents for problem-solving

Solving exponential equations

• Use the properties of exponents to simplify expressions and solve equations involving exponents
• Solve exponential equations by using logarithms to isolate the variable in the exponent
  • Example: Solve $2^x = 32$
    • $\log_2(2^x) = \log_2(32)$
    • $x \log_2(2) = \log_2(32)$
    • $x \cdot 1 = 5$
    • $x = 5$

Modeling with exponential functions

• Apply exponential functions to model real-world situations involving exponential growth or decay (population growth, radioactive decay, compound interest)
• Understand and apply the continuous growth (or decay) formula $A = A_0e^{kt}$, where $A$ is the final amount, $A_0$ is the initial amount, $e$ is the mathematical constant (approximately 2.71828), $k$ is the continuous growth (or decay) rate, and $t$ is the time
  • Example: A population of bacteria starts with 100 cells and doubles every 2 hours. How many bacteria will be present after 6 hours?
    • $A = 100e^{kt}$, where $k = \frac{\ln(2)}{2} \approx 0.347$ and $t = 6$
    • $A = 100e^{0.347 \cdot 6} \approx 800$ bacteria
• Recognize and solve problems involving exponential functions in various contexts (biology, physics, economics, computer science)