Exponential Graphs

An exponential function is typically written in the form $y = ab^x$ or $y = ab^{-x}$, where $a$ and $b$ are constants and $b > 0$.

The variable $x$ is the exponent, which distinguishes these functions from power functions like $y = x^2$ where the variable is the base.

The constant $a$ represents the y-intercept of the graph because when $x = 0$, $y = a \cdot b^0 = a \cdot 1 = a$.

The base $b$ determines the rate of growth or decay; it must be positive to ensure the function is defined for all real numbers.

Growth vs. Decay: If the base $b > 1$, the function exhibits exponential growth, increasing more rapidly as $x$ increases. If $0 < b < 1$, the function exhibits exponential decay, approaching zero as $x$ increases.

Horizontal Asymptote: The x-axis ($y = 0$) acts as a horizontal asymptote for the basic form $y = ab^x$. This means the curve gets infinitely close to the x-axis but never actually touches or crosses it.

Non-Linearity: Unlike linear functions that have a constant slope, exponential functions have a slope that changes at every point, becoming steeper in growth and flatter in decay.

Domain and Range: For $y = ab^x$ (where $a > 0$), the domain is all real numbers ($-\infty < x < \infty$), and the range is all positive real numbers ($y > 0$).

Finding the Equation by Inspection: If a graph is provided, the value of $a$ can often be identified immediately as the y-coordinate where the curve crosses the y-axis.

Solving for the Base: To find $b$, substitute a known coordinate $(x, y)$ from the curve into the equation $y = ab^x$ and solve for $b$ using algebraic methods or logarithms.

Plotting the Curve: When sketching, start by marking the y-intercept, then calculate 1-2 points for positive $x$ and 1-2 points for negative $x$ to determine the steepness.

Handling Reflections: If the coefficient $a$ is negative, the entire graph is reflected across the x-axis, changing the range to $y < 0$.

Exponential vs. Power Functions: In $y = 2^x$ (exponential), the rate of growth increases with $x$. In $y = x^2$ (power), the base is the variable. Exponential growth eventually surpasses any polynomial growth.

Positive vs. Negative $x$: For growth, negative $x$ values result in small fractions (approaching the asymptote), while positive $x$ values result in large numbers.

Check the Intercept First: Always look at the y-axis. If the curve passes through $(0, 1)$, then $a = 1$. If it passes through $(0, 5)$, then $a = 5$.

Asymptote Awareness: Never draw the curve crossing the horizontal asymptote. In exams, ensure your sketch clearly shows the curve leveling off near the x-axis (or the shifted asymptote).

Substitution Accuracy: When solving for $b$ using a point like $(2, 16)$ and $a=1$, remember that $16 = b^2$ implies $b = 4$. Always check if $b$ must be positive (which it must for standard exponential functions).

Verify Growth/Decay: Before finalizing an equation, check if your $b$ value matches the graph's direction. If the graph is falling but your $b$ is greater than 1, you likely have a sign error in the exponent.

The 'Zero' Trap: Students often think $b^0 = 0$. Remember that any non-zero base raised to the power of zero is 1, which is why the basic intercept is $(0, 1)$.

Negative Bases: A common mistake is attempting to use a negative base (e.g., $y = (-2)^x$). This does not produce a continuous curve and is not considered a standard exponential function.

Confusing Shifts with Bases: A vertical shift (e.g., $y = 2^x + 3$) moves the horizontal asymptote to $y = 3$. Students often forget to adjust the asymptote when the function is shifted.