An exponential graph represents an exponential function, which has the general form:

$y = a \cdot b^x$

where:

• $a$ is the initial value or the $y$-intercept (when $x = 0$),
• $b$ is the base of the exponential, which determines the growth (if $b > 1$) or decay (if $0 < b < 1$) rate,
• $x$ is the exponent.

Key Characteristics of an Exponential Graph:

1. Shape: The graph is a curve that either rises rapidly (if $b > 1$) or falls rapidly (if $0 < b < 1$).
2. Intercept: The $y$-intercept occurs at $(0, a)$.
3. Asymptote: The graph has a horizontal asymptote, usually along the x-axis ($y = 0$) if $a$ is positive.
4. Domain and Range:
   • Domain: The domain is all real numbers, $x \in (-\infty, \infty)$.
   • Range: The range is $y > 0$ if $a > 0$ and $b > 0$, or $y < 0$ if $a < 0$.

Example:

For the function $y = 2 \cdot 3^x$:

• The base $b = 3$ indicates the graph will grow exponentially.
• The initial value $a = 2$ means the $y$-intercept is at $(0, 2)$.
• The graph will rapidly increase as $x$ becomes positive and approach zero as $x$ becomes negative.

Would you like to see a specific exponential graph or have any questions?

Here are 5 related questions to expand on this:

1. How does the value of $b$ affect the steepness of an exponential graph?
2. What happens to the graph if the base $b$ is between 0 and 1?
3. How do you shift an exponential graph vertically or horizontally?
4. How do you find the inverse of an exponential function?
5. What is the significance of the horizontal asymptote in an exponential graph?

Tip: When graphing an exponential function, always identify the $y$-intercept and asymptote first to guide your sketch.