Find the exponential function f(x)=C a^x whose graph is...

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Key Concepts

Exponential Functions

An exponential function is a mathematical function of the form f(x) = C · a^x, where C and a are constants. This type of function exhibits constant proportional change, meaning that for each unit increase in x, the function is multiplied by the factor a. Exponential functions are commonly used to model growth or decay processes.

Parameters of Exponential Functions

In the function f(x) = C · a^x, the constant C represents the initial value or the y-intercept when x = 0, and it scales the overall function. The base a determines the rate of change of the function: if a > 1, the function represents exponential growth, and if 0 < a < 1, it represents exponential decay. These parameters are crucial in defining the behavior of the graph.

Graphical Analysis for Parameter Determination

Graphical analysis involves examining key points on the graph of the exponential function, such as the y-intercept and another point on the curve, to determine the values of C and a. By reading the value of the function at x = 0, one can directly find C. Using another point on the graph and substituting into the equation allows solving for the base a. This process helps in reconstructing the specific exponential function from its graph.

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