6.2 Graphs of Exponential Functions

Graphing Exponential Functions

Exponential functions model situations where quantities grow or shrink by a constant factor over equal intervals. Think of a bank account earning compound interest (growth) or a medication leaving your bloodstream (decay). The graphs of these functions have a distinctive curved shape with predictable features you can identify from the equation alone.

This section covers the key features of exponential graphs, how transformations change their shape and position, and how to compare different exponential functions.

Key Features of Exponential Graphs

The general form of an exponential function is $f(x) = ab^x$, where $a$ and $b$ each control different aspects of the graph.

The coefficient $a$ controls vertical stretch, compression, and reflection:

• $|a| > 1$ vertically stretches the graph (pulls it away from the x-axis)
• $0 < |a| < 1$ vertically compresses the graph (pushes it toward the x-axis)
• $a < 0$ reflects the graph across the x-axis, flipping it upside down

The base $b$ determines whether the function grows or decays:

• $b > 1$: exponential growth (e.g., $f(x) = 2^x$ doubles for every increase of 1 in $x$)
• $0 < b < 1$: exponential decay (e.g., $f(x) = (0.5)^x$ halves for every increase of 1 in $x$)

Note that $b$ must be positive and cannot equal 1. If $b = 1$, the function is just the constant $f(x) = a$, which isn't exponential.

Y-intercept: Set $x = 0$ and evaluate. Since $b^0 = 1$, the y-intercept of $f(x) = ab^x$ is always $(0, a)$.

Horizontal asymptote: For the basic form $f(x) = ab^x$, the horizontal asymptote is $y = 0$ (the x-axis). The graph gets closer and closer to this line but never touches it.

End behavior depends on whether you have growth or decay (assuming $a > 0$):

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

In both cases, the graph approaches the asymptote on one side and rises steeply on the other.

Transformations of Exponential Functions

The full transformed form is $f(x) = ab^{x-h} + k$. Each parameter shifts or reshapes the graph in a specific way.

Vertical shift (the $k$ value): Moves the entire graph up or down.

• $k > 0$ shifts the graph up by $k$ units
• $k < 0$ shifts the graph down by $|k|$ units
• This also moves the horizontal asymptote from $y = 0$ to $y = k$

Horizontal shift (the $h$ value): Moves the graph left or right.

• $h > 0$ shifts the graph right by $h$ units
• $h < 0$ shifts the graph left by $|h|$ units
• Watch the sign carefully: $f(x) = 2^{x-3}$ shifts right 3, while $f(x) = 2^{x+3}$ shifts left 3

Vertical stretch/compression (the $a$ value): Covered above. Larger $|a|$ makes the curve steeper; smaller $|a|$ flattens it.

Reflections:

• Across the x-axis: Make $a$ negative. For example, $f(x) = -2^x$ flips the growth curve below the x-axis.
• Across the y-axis: Negate the input. $f(x) = b^{-x}$ is equivalent to $f(x) = \left(\frac{1}{b}\right)^x$, which turns growth into decay and vice versa.

Analyzing and Comparing Exponential Functions

Comparison of Exponential Graphs

When you're asked to compare two exponential functions, focus on these features:

• Effect of changing $a$: A larger $|a|$ makes the graph steeper (it rises or falls faster). Changing the sign of $a$ reflects the graph across the x-axis.
• Effect of changing $b$: Among growth functions ($b > 1$), a larger base means faster growth. For example, $3^x$ grows faster than $2^x$. Among decay functions ($0 < b < 1$), a base closer to 0 means faster decay.
• Y-intercepts: The y-intercept is $(0, a + k)$ in the transformed form. Two functions with different $a$ or $k$ values will cross the y-axis at different heights.
• Asymptotes: Functions with the same $k$ value share the same horizontal asymptote $y = k$. Without a vertical shift, all basic exponential functions share the asymptote $y = 0$.
• End behavior: Functions with the same type of base (both $b > 1$ or both $0 < b < 1$) have the same general end behavior, though they may grow or decay at different rates.

Properties of Exponential Functions

• Domain: All real numbers. You can plug any value of $x$ into an exponential function.
• Range: For $f(x) = ab^x + k$ with $a > 0$, the range is $(k, \infty)$. With $a < 0$, the range is $(-\infty, k)$. The output never equals $k$ because the graph never touches the asymptote.
• Continuity: Exponential functions are smooth and unbroken, with no gaps, holes, or jumps.
• One-to-one: Every output corresponds to exactly one input, which is why exponential functions have inverses (logarithms).
• Natural exponential function: The special case where the base is $e \approx 2.71828$. The function $f(x) = e^x$ appears frequently in calculus and in continuous growth/decay models like $A = Pe^{rt}$.