Exponential functions are powerful tools for modeling growth and decay. They have a unique shape, with a y-intercept, horizontal asymptote, and distinct end behavior. Understanding these key features helps us analyze real-world scenarios like population growth or radioactive decay.

Transformations of exponential functions allow us to shift, stretch, or reflect their graphs. By adjusting parameters, we can model various situations, from delayed growth to accelerated decay. Comparing different exponential functions helps us understand their relative behaviors and applications.

Graphing Exponential Functions

Key features of exponential graphs

• General form of exponential functions: $f(x) = ab^x$
  • $a$ represents vertical stretch or compression and reflection across y-axis
    • $|a| > 1$ vertically stretches the graph
    • $0 < |a| < 1$ vertically compresses the graph
    • $a < 0$ reflects the graph across x-axis
  • $b$ represents the base and determines growth or decay
    • $b > 1$ indicates exponential growth (doubles every day)
    • $0 < b < 1$ indicates exponential decay (half-life of radioactive elements)
• y-intercept: point where the graph crosses y-axis found by evaluating $f(0)$
• Horizontal asymptote: line the graph approaches as $x$ approaches positive or negative infinity
  • Exponential growth functions have horizontal asymptote $y = 0$ as $x \to -\infty$
  • Exponential decay functions have horizontal asymptote $y = 0$ as $x \to \infty$
• End behavior: trend of the graph as $x$ approaches positive or negative infinity
  • Exponential growth functions have $f(x) \to \infty$ as $x \to \infty$ and $f(x) \to 0$ as $x \to -\infty$ (population growth)
  • Exponential decay functions have $f(x) \to 0$ as $x \to \infty$ and $f(x) \to \infty$ as $x \to -\infty$ (medication concentration in body)

Transformations of exponential functions

• Vertical shift: moves graph up or down by adding or subtracting constant $k$ to function
  • $f(x) = ab^x + k$ shifts graph vertically by $k$ units
    • $k > 0$ shifts graph up (increased initial investment)
    • $k < 0$ shifts graph down (lower starting population)
• Horizontal shift: moves graph left or right by adding or subtracting constant $h$ to input variable $x$
  • $f(x) = ab^{x-h}$ shifts graph horizontally by $h$ units
    • $h > 0$ shifts graph right (delayed start of exponential growth)
    • $h < 0$ shifts graph left (earlier onset of exponential decay)
• Vertical stretch or compression: multiplies output values by constant $a$ affecting steepness of graph
  • $|a| > 1$ stretches graph vertically (faster growth rate)
  • $0 < |a| < 1$ compresses graph vertically (slower decay rate)
• Reflection: flips graph across x-axis or y-axis
  • Reflection across x-axis occurs when $a < 0$ in general form
  • Reflection across y-axis occurs when input variable $x$ is negated $f(x) = ab^{-x}$

Analyzing and Comparing Exponential Functions

Comparison of exponential graphs

• Changes in $a$:
  • Increasing $|a|$ makes graph steeper while decreasing $|a|$ makes it less steep (comparing growth rates)
  • Changing sign of $a$ reflects graph across x-axis (growth vs decay)
• Changes in $b$:
  • Increasing $b$ ($b > 1$) makes graph grow faster while decreasing $b$ ($0 < b < 1$) makes graph decay slower (base represents growth/decay factor)
• Comparing y-intercepts:
  • y-intercept affected by value of $a$ and any vertical shift $k$ (different initial values)
• Comparing asymptotes:
  • Exponential functions with same base $b$ share same horizontal asymptote $y = 0$ (common long-term behavior)
• Comparing end behavior:
  • Functions with same base $b$ exhibit similar end behavior depending on $b > 1$ (growth) or $0 < b < 1$ (decay)

Properties of Exponential Functions

• Domain: All real numbers, as exponential functions are defined for every input value
• Range: All positive real numbers for exponential functions with positive base (excluding zero)
• Continuous function: Exponential functions are smooth and unbroken, with no gaps or jumps in their graphs
• Exponential equation: An equation where the variable appears in the exponent (e.g., $2^x = 8$)
• Natural exponential function: A special case where the base is the mathematical constant e (approximately 2.71828)