Exponential Functions

• An exponential function is defined by the form $y = a^x$, where the base $a$ is a positive constant ($a > 0$) and the exponent $x$ is the independent variable.

• The domain of an exponential function is all real numbers, while the range is strictly positive ($y > 0$) because a positive base raised to any power can never result in zero or a negative number.

• Every basic exponential graph of the form $y = a^x$ passes through the y-intercept $(0, 1)$, as any non-zero number raised to the power of zero equals one ($a^0 = 1$).

• Asymptotic Behavior: The x-axis ($y = 0$) acts as a horizontal asymptote for all basic exponential functions. As $x$ moves toward negative infinity (for growth) or positive infinity (for decay), the curve approaches but never touches the x-axis.

• Growth vs. Decay: If the base $a > 1$, the function exhibits exponential growth, meaning the values increase at an accelerating rate. If $0 < a < 1$, the function exhibits exponential decay, where values decrease toward zero.

• The Natural Base $e$: The number $e$ (approximately $2.718$) is a unique irrational constant. It is the only base where the value of the function $y = e^x$ is exactly equal to the gradient (slope) of the curve at any given point $x$.

Sketching Exponential Graphs

• Identify the Intercept: Start by plotting the point $(0, 1)$ unless a vertical translation or stretch is present.
• Determine the Direction: Check if the base $a$ is greater than 1 (upward slope to the right) or between 0 and 1 (downward slope to the right).
• Draw the Asymptote: Clearly indicate the horizontal asymptote, usually the x-axis, and ensure the curve approaches it without crossing.

Differentiation of Exponential Functions

• The Basic Rule: The derivative of the natural exponential function is itself: $\frac{d}{dx}(e^x) = e^x$.
• The Chain Rule Application: For functions of the form $y = e^{kx}$, the derivative is found by multiplying the function by the constant $k$: $\frac{d}{dx}(e^{kx}) = ke^{kx}$
• Negative Exponents: For decay functions like $y = e^{-kx}$, the derivative is $\frac{d}{dx}(e^{-kx}) = -ke^{-kx}$, indicating a negative gradient throughout.

• It is vital to distinguish between exponential functions ($y = a^x$) and power functions ($y = x^a$). In exponential functions, the variable is in the exponent, leading to much faster growth rates than power functions where the variable is the base.

• Reflections: The graph of $y = a^{-x}$ is a reflection of $y = a^x$ in the y-axis. This transformation effectively turns a growth function into a decay function with the reciprocal base $y = (\frac{1}{a})^x$.

• Labeling: When sketching, always explicitly label the y-intercept and the equation of the horizontal asymptote (e.g., $y=0$). Examiners look for these specific features to award marks.

• Exact Values: If a question asks for a gradient or a coordinate at a specific point, provide the answer in terms of $e$ (e.g., $3e^2$) unless a decimal approximation is specifically requested.

• Steepness Comparison: When asked to sketch two growth functions on the same axes (e.g., $y=2^x$ and $y=5^x$), ensure the one with the larger base is steeper for $x > 0$ and closer to the x-axis for $x < 0$.

• Sanity Check: Remember that $e^x$ can never be negative. If you solve an equation and get $e^x = -2$, there are no real solutions.

• The Zero Power Error: A common mistake is assuming $a^0 = 0$. Always remember that any base to the power of zero is 1, which is why the y-intercept is $(0,1)$.

• Negative Bases: Students sometimes attempt to use negative numbers as the base $a$. This is not allowed in standard exponential functions because it would result in non-real numbers for fractional exponents (e.g., $(-4)^{0.5} = \sqrt{-4}$).

• Chain Rule Neglect: When differentiating $e^{5x}$, students often forget to multiply by the 5, incorrectly stating the derivative is just $e^{5x}$.