In Exercises 19–24, the graph of an exponential function is given. Select the function for each graph from the following options: f(x) = 3^x, g(x) = 3^(x-1), h(x) = 3^x - 1 ; f(x) = -3^x, G(x) = 3^(-x), H(x) = -3^(-x)

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1. Observe the graph and note that it is decreasing as x increases, indicating a negative exponent or a reflection.
2. Identify that the graph approaches the x-axis as x goes to positive infinity, suggesting a horizontal asymptote at y = 0.
3. Check the y-intercept of the graph, which is at (0,1), consistent with functions of the form 3^(-x).
4. Consider the options: f(x) = 3^x, g(x) = 3^(x-1), h(x) = 3^x - 1, f(x) = -3^x, G(x) = 3^(-x), H(x) = -3^(-x).
5. Select G(x) = 3^(-x) as it matches the observed behavior of the graph.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. These functions exhibit rapid growth or decay, depending on whether the base is greater than or less than one. Understanding their general shape and behavior is crucial for analyzing their graphs.

Graph Characteristics

The graph of an exponential function typically features a horizontal asymptote, which is a line that the graph approaches but never touches. For functions of the form f(x) = b^x (where b > 1), the graph rises steeply to the right and approaches zero to the left. Conversely, for functions like g(x) = b^(-x), the graph decreases as x increases, illustrating the concept of exponential decay.

Transformations of Functions

Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For example, the function g(x) = 3^(x-1) represents a horizontal shift of the base function f(x) = 3^x to the right by one unit. Understanding these transformations helps in predicting how changes in the function's equation affect its graph.