Transformations and Analysis of Exponential Functions

Exponential and Logarithmic Functions

Transformations of Exponential Functions

Exponential functions are a key topic in College Algebra, often written in the form y = a \, b^{x} + c. Understanding how to graph and analyze these functions involves applying transformations to the basic exponential graph.

• Key Point 1: The parent function for exponentials is y = e^{x} or y = a^{x}, where a > 0 and a \neq 1.

• Key Point 2: Transformations include vertical shifts, horizontal shifts, reflections, and stretches/compressions.

Example: Graphing $y = e^{x} + 1$

• Step 1: Start with the parent function $y = e^{x}$.

• Step 2: The '+1' indicates a vertical shift upward by 1 unit. The graph of $y = e^{x}$ is moved up so that every point $(x, y)$ becomes $(x, y+1)$.

• Step 3: The domain of $y = e^{x} + 1$ is all real numbers: $(-\infty, \infty)$.

• Step 4: The range is $(1, \infty)$, since $e^{x}$ is always positive and the minimum value is shifted up by 1.

• Step 5: The horizontal asymptote is $y = 1$.

Table: Transformations of $y = e^{x}$

Transformation	Equation	Effect on Graph
Vertical Shift Up	$y = e^{x} + k$	Moves graph up by $k$ units
Vertical Shift Down	$y = e^{x} - k$	Moves graph down by $k$ units
Horizontal Shift Right	$y = e^{x-h}$	Moves graph right by $h$ units
Horizontal Shift Left	$y = e^{x+h}$	Moves graph left by $h$ units
Reflection over x-axis	$y = -e^{x}$	Flips graph upside down

Domain, Range, and Asymptotes

Analyzing the domain, range, and asymptotes is essential for understanding exponential functions.

• Domain: For $y = e^{x} + 1$, the domain is all real numbers: $(-\infty, \infty)$.

• Range: Since $e^{x}$ is always positive, $y = e^{x} + 1$ is always greater than 1. Thus, the range is $(1, \infty)$.

• Horizontal Asymptote: As $x \to -\infty$, $e^{x} \to 0$, so $y \to 1$. The horizontal asymptote is $y = 1$.

Example:

• Given $y = e^{x} + 1$, the graph is the standard exponential curve shifted up by 1 unit.

• Domain: $(-\infty, \infty)$

• Range: $(1, \infty)$

• Horizontal Asymptote: $y = 1$

Additional info: The original file included a graph and asked for transformations, domain, range, and asymptote. The above notes expand on these concepts for clarity and completeness.