Exponential Functions and Graphs

An exponential function is a mathematical function where the variable is located in the exponent. The general form is:

$y = ab^x$

Where:

• $a$ is the initial value (or y-intercept when there are no horizontal shifts).
• $b$ is the base, a positive constant ($b > 0$) not equal to 1 ($b \neq 1$).
• $x$ is the independent variable.

Exponential Growth vs. Decay

The base $b$ determines whether the graph represents exponential growth or decay:

• Exponential Growth ($b > 1$): As $x$ increases, $y$ increases rapidly. The graph curves upward, moving away from the x-axis. Example: $y = 2^x$ or $y = 3^x$.
• Exponential Decay ($0 < b < 1$): As $x$ increases, $y$ decreases, getting closer and closer to zero. The graph curves downward, approaching the x-axis. Example: $y = (\frac{1}{2})^x$.

For the basic parent function $y = b^x$, the x-axis ($y = 0$) acts as a horizontal asymptote—a horizontal line the graph approaches but never actually touches.

The Natural Base $e$

In many higher-level math and real-world applications (like continuous compound interest or population growth), you will encounter the natural base, denoted by $e$.

$e \approx 2.71828...$

Because $e > 1$, the function $y = e^x$ is an exponential growth function. It follows all the exact same graphing rules as any other base greater than 1.

Transformations of Exponential Graphs

We can shift, stretch, and reflect exponential graphs using the standard transformation formula:

$y = a \cdot b^{x - h} + k$

• $a$ (Vertical Stretch/Compression): If $|a| > 1$, the graph stretches vertically. If $0 < |a| < 1$, it compresses. If $a$ is negative, the graph reflects upside down across the horizontal asymptote.
• $h$ (Horizontal Shift): Moves the graph left or right. (e.g., $x - 2$ shifts right by 2; $x + 3$ shifts left by 3).
• $k$ (Vertical Shift): Moves the graph up or down. Crucially, this changes the horizontal asymptote to $y = k$.

Example: Graphing a Transformation

Let's analyze and graph the function:

$y = 2 \cdot 3^{x - 1} - 4$

1. Identify the Parent Function: The base is $3$, so the parent function is $y = 3^x$ (exponential growth).
2. Find the Horizontal Asymptote: The vertical shift is $k = -4$, so the horizontal asymptote is the line $y = -4$.
3. Apply Transformations:
   • The $x - 1$ in the exponent shifts the graph right by 1.
   • The $2$ in front stretches the graph vertically by a factor of 2.
   • The $- 4$ shifts the graph down by 4.
4. Plot Key Points: Choose easy $x$-values to find exact coordinates.
   • Let $x = 1$: $y = 2 \cdot 3^{1 - 1} - 4 = 2(3^0) - 4 = 2(1) - 4 = -2$. Point: $(1, -2)$.
   • Let $x = 2$: $y = 2 \cdot 3^{2 - 1} - 4 = 2(3^1) - 4 = 6 - 4 = 2$. Point: $(2, 2)$.

To sketch, draw the dotted line $y = -4$, plot the points $(1, -2)$ and $(2, 2)$, and draw a smooth curve starting near the asymptote on the left and growing steeply upward to the right.