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Exponential functions are a critical part of many mathematical and real-world applications. Fundamentally, these functions have the form \( f(x) = a \cdot b^x \), where \( a \) is a constant, \( b \) is the base of the exponential, and \( x \) is the exponent. In the function \( f(x) = e^x \), the number "e" (approximately 2.718) is a unique base, known as Euler's number, which arises naturally in various mathematical fields.
Understanding how these functions behave requires a grasp of their unique properties:

• Exponential functions exhibit rapid growth, meaning they increase very quickly compared to other types of functions like linear or quadratic functions.
• Their graphs are characterized by a constant percentage rate of growth.
• They continuously grow faster as \( x \) increases, leading to steep upward curves on a graph.
• They never touch the x-axis, always remaining positive if the base is greater than 1.

This behavior makes exponential functions suitable for modeling situations of continual increase, such as compound interest, population growth, and even certain biological processes.

Vertical stretching is a type of transformation applied to a function which affects the graph's appearance without altering its shape. It is performed by multiplying the function by a constant factor (greater than 1) which expands or compresses the graph along the y-axis.
For example, consider the function \( f(x) = e^x \). If we transform it by multiplying the entire function by a factor of \( \frac{4}{3} \) to get \( g(x) = \frac{4}{3}e^x \), this indicates a vertical stretch. Here’s what happens with vertical stretching:

• The y-values of the graph are multiplied by the stretching factor.
• This makes the graph appear taller, emphasizing its steepness as it rises.
• The y-intercept is also affected; for instance, a vertical stretch would change the y-intercept of \( e^x \) from 1 to \( \frac{4}{3} \) for \( \frac{4}{3}e^x \).
• The x-values remain unchanged, preserving the horizontal position of each point on the graph.

Vertical stretching is significant as it allows us to adjust the intensity of growth or decline in models using exponential functions.

Graphing functions is a vital skill in mathematics, allowing for a visual representation of how a function behaves. When graphing exponential functions like \( f(x) = e^x \) and its vertically stretched version \( g(x) = \frac{4}{3}e^x \), certain steps must be followed to ensure accuracy:

• Identify key points on the initial function, such as the y-intercept and one or two additional points, to understand the curve’s growth.
• Understand the asymptotic behavior. The graph of an exponential function approaches but never touches the x-axis.
• Plot these key points on a graph, drawing a smooth curve through them which represents the exponential growth.
• Apply any transformations (like vertical stretching) by adjusting the y-values of these points according to the stretching factor.
• Re-draw the graph with these adjusted values, ensuring that the curve retains its smooth shape while showing the modified steepness or height.

Through this method, you can correctly plot both the original function and its transformed versions, providing clear insights into how exponential functions can be manipulated visually through graphing.