To create a table that demonstrates the growth characteristics of exponential functions, Romona should follow these steps:
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Choose an Exponential Function: Select a specific exponential function to analyze, such as \( f(x) = a \cdot b^x \) where \( a \) is a positive constant and \( b > 1 \). A common choice could be \( f(x) = 2^x \) or \( f(x) = 3^x \).
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Select Input Values: Choose a range of input values (x-values) for which she will compute the corresponding output values. It's beneficial to use both positive and negative integers. For example, Romona might select values like -2, -1, 0, 1, 2, 3, etc.
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Calculate Output Values: For each input value, calculate the output using the chosen exponential function. Create a simple table, like this:
| x | f(x) | |----|--------| | -2 | 1/4 | | -1 | 1/2 | | 0 | 1 | | 1 | 2 | | 2 | 4 | | 3 | 8 | | 4 | 16 |
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Analyze the Growth: After constructing the table, Romona should analyze how quickly the output values increase as x increases. She can look at how the outputs double (or increase by a certain factor) as she moves from one integer value of x to the next.
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Compare with Linear Functions: To further highlight the growth of exponential functions, Romona can create a similar table for a linear function, such as \( g(x) = 2x \), and compare the values in the two tables.
What Romona Can Prove
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Rapid Growth: Through her table, Romona will be able to demonstrate that exponential functions grow significantly faster than linear functions. She will observe that after a certain point (typically around x = 6 or 7), the values of the exponential function far exceed those of the linear function.
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Doubling Nature: She can also prove that for any exponential function of the form \( f(x) = a \cdot b^x \) (where \( b > 1 \)), as x increases by 1, the output roughly doubles (if \( b = 2\)) or increases by a fixed multiplicative factor (if \( b \) is a constant greater than 1).
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Behavior for Negative Values: The table will also illustrate that for negative x-values, the outputs of exponential functions approach 0 but never actually reach it, demonstrating asymptotic behavior.
Ultimately, Romona can use her table to effectively illustrate and prove the unique and rapid growth characteristics of exponential functions compared to other types of functions.