Exponential functions model growth and decay, with key features like y-intercepts and asymptotes. Their graphs are shaped by the base and coefficients, determining steepness and direction. Understanding these elements helps visualize and interpret exponential relationships.

Transformations like shifts, stretches, and reflections alter exponential graphs while preserving their core shape. These changes affect y-intercepts and asymptotes. Analyzing these impacts helps predict how adjustments to function parameters influence graph behavior and real-world applications.

Graphing Exponential Functions

Key features of exponential graphs

• General form of an exponential function $f(x) = ab^x$
  • $a$ represents the vertical stretch factor and y-intercept at the point $(0, a)$
  • $b$ represents the base and determines whether the function exhibits growth or decay
    • Function exhibits exponential growth when $b > 1$ (doubles every day)
    • Function exhibits exponential decay when $0 < b < 1$ (half-life of radioactive elements)
• Horizontal asymptote at $y = 0$ for both growth and decay functions
• End behavior differs for exponential growth and decay
  • Exponential growth ($b > 1$) approaches infinity as $x$ approaches infinity, and approaches 0 as $x$ approaches negative infinity
  • Exponential decay ($0 < b < 1$) approaches 0 as $x$ approaches infinity, and approaches infinity as $x$ approaches negative infinity
• Exponential functions are continuous functions, meaning they have no breaks or gaps in their graphs

Transformations of exponential functions

• Vertical shift of $f(x) = ab^x + k$ moves the graph up by $k$ units when $k > 0$ or down by $|k|$ units when $k < 0$
  • Vertical shift changes the y-intercept to $(0, a + k)$
  • Horizontal asymptote becomes $y = k$
• Horizontal shift of $f(x) = ab^{x - h}$ moves the graph right by $h$ units when $h > 0$ or left by $|h|$ units when $h < 0$
  • Horizontal shift changes the y-intercept to $(h, a)$
• Vertical stretch/compression of $f(x) = cab^x$ stretches the graph vertically by a factor of $|c|$ when $|c| > 1$ or compresses the graph vertically by a factor of $|c|$ when $0 < |c| < 1$
  • Vertical stretch/compression changes the y-intercept to $(0, ca)$
• Reflection across the x-axis of $f(x) = -ab^x$ reflects the graph over the x-axis
  • Reflection across the x-axis changes the y-intercept to $(0, -a)$
• Reflection across the y-axis of $f(x) = ab^{-x}$ reflects the graph over the y-axis
  • Reflection across the y-axis does not change the y-intercept $(0, a)$

Analyzing Exponential Functions

Effects of base and coefficients

• Changing the base $b$ affects the steepness of the growth or decay curve
  • For $b > 1$, increasing $b$ results in a steeper growth curve, while decreasing $b$ results in a less steep growth curve (comparing $2^x$ and $1.5^x$)
  • For $0 < b < 1$, increasing $b$ results in a less steep decay curve, while decreasing $b$ results in a steeper decay curve (comparing $0.5^x$ and $0.8^x$)
• Changing the coefficient $a$ affects the vertical stretch/compression and reflection across the x-axis
  • Increasing $|a|$ vertically stretches the graph by a factor of $|a|$, while decreasing $|a|$ vertically compresses the graph by a factor of $|a|$ (comparing $2^x$ and $3 \cdot 2^x$)
  • If $a < 0$, the graph is reflected across the x-axis (comparing $2^x$ and $-2^x$)
• Changing the coefficient $c$ in $f(x) = cab^x$ affects the vertical stretch/compression and reflection across the x-axis
  • Increasing $|c|$ vertically stretches the graph by a factor of $|c|$, while decreasing $|c|$ vertically compresses the graph by a factor of $|c|$ (comparing $2^x$ and $2 \cdot 2^x$)
  • If $c < 0$, the graph is reflected across the x-axis (comparing $2^x$ and $-2 \cdot 2^x$)

Related Functions and Models

• Exponential models are used to describe real-world phenomena with exponential growth or decay patterns
• Logarithmic functions are the inverse functions of exponential functions
  • The natural logarithm, denoted as ln(x), is the inverse of the exponential function with base e (Euler's number)