4.2 Graphs of Exponential Functions

Graphing Exponential Functions

Exponential functions model situations where a quantity multiplies by a constant factor over equal intervals. Their graphs have a distinctive curved shape controlled by the base, coefficients, and transformations, so reading these features helps you connect equations, graphs, and real-world models.

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Key Features of Exponential Graphs

The general form of an exponential function is $f(x) = ab^x$, where each parameter plays a specific role:

• $a$ is the initial value (the y-intercept). When you plug in $x = 0$, you get $f(0) = a$, so the graph always passes through $(0, a)$.
• $b$ is the base, and it controls whether the function grows or decays:
  • Exponential growth occurs when $b > 1$. The output increases as $x$ increases. Think of a population doubling every year: $f(x) = 100 \cdot 2^x$.
  • Exponential decay occurs when $0 < b < 1$. The output decreases as $x$ increases. Radioactive half-life is a classic example: $f(x) = 500 \cdot 0.5^x$.
• The base $b$ must be positive and cannot equal 1 (since $1^x = 1$ for all $x$, which is just a constant).

Horizontal asymptote: For the parent function $f(x) = ab^x$, the horizontal asymptote is $y = 0$. The graph approaches the x-axis but never touches it.

End behavior depends on whether you have growth or decay:

• Growth ($b > 1$): As $x \to \infty$, $f(x) \to \infty$. As $x \to -\infty$, $f(x) \to 0$.
• Decay ($0 < b < 1$): As $x \to \infty$, $f(x) \to 0$. As $x \to -\infty$, $f(x) \to \infty$.

Exponential functions are continuous, meaning their graphs have no breaks, holes, or jumps. They also have a domain of all real numbers and a range of $(0, \infty)$ for the parent form (assuming $a > 0$).

Transformations of Exponential Functions

Transformations follow the same rules you've seen with other function families. The general transformed form is:

$f(x) = a \cdot b^{x - h} + k$

Here's how each piece affects the graph:

• Vertical shift ($k$): Moves the graph up ($k > 0$) or down ($k < 0$).
  • The horizontal asymptote shifts to $y = k$
  • The y-intercept becomes $(0, a \cdot b^{-h} + k)$
• Horizontal shift ($h$): Moves the graph right ($h > 0$) or left ($h < 0$).
  • The horizontal asymptote is unaffected
  • The point that was the y-intercept $(0, a)$ shifts to $(h, a)$, but the actual y-intercept (where the graph crosses $x = 0$) is found by evaluating $f(0) = a \cdot b^{-h} + k$
• Vertical stretch/compression (multiplying by $c$): In $f(x) = c \cdot ab^x$, the factor $|c|$ stretches the graph vertically when $|c| > 1$ and compresses it when $0 < |c| < 1$.
  • The y-intercept becomes $(0, ca)$
• Reflection across the x-axis: $f(x) = -ab^x$ flips the graph upside down.
  • The y-intercept becomes $(0, -a)$
  • The range flips to $(-\infty, 0)$ for the parent form
• Reflection across the y-axis: $f(x) = ab^{-x}$ mirrors the graph left-to-right.
  • The y-intercept stays at $(0, a)$ since $b^{-0} = 1$
  • This effectively converts growth into decay and vice versa (since $b^{-x} = (1/b)^x$)

Analyzing Exponential Functions

Effects of Base and Coefficients

The base $b$ controls the rate of growth or decay, while the coefficient controls the scale.

Changing the base:

• For growth functions ($b > 1$), a larger base means steeper growth. Compare $3^x$ (triples each step) versus $1.5^x$ (increases by 50% each step).
• For decay functions ($0 < b < 1$), a base closer to 0 means steeper decay. Compare $0.2^x$ (drops to 20% each step) versus $0.8^x$ (drops to 80% each step).

Changing the coefficient $a$:

• Increasing $|a|$ stretches the graph vertically. For example, $3 \cdot 2^x$ is three times as tall as $2^x$ at every point.
• Making $a$ negative reflects the graph across the x-axis. So $-2^x$ is the mirror image of $2^x$ flipped over the x-axis.

Note that in the form $f(x) = cab^x$, the product $ca$ acts as a single coefficient. Writing $f(x) = 6 \cdot 2^x$ is the same whether you think of it as $c = 3, a = 2$ or $c = 1, a = 6$. What matters for graphing is the overall coefficient in front of $b^x$.

Related Functions and Models

Exponential models show up in compound interest, population growth, radioactive decay, and cooling/heating problems. You'll work with these applications throughout the unit.

Logarithmic functions are the inverses of exponential functions. If $y = b^x$, then $x = \log_b(y)$. The natural logarithm $\ln(x)$ is the inverse of $e^x$, where $e \approx 2.718$ is Euler's number. You'll explore logarithms in detail in the next section, but for now, know that the graph of a logarithmic function is the reflection of its corresponding exponential function across the line $y = x$.