Given the tables below, determine which function represents an exponential growth.
Understand the Problem
The question is asking to identify a function that demonstrates exponential growth based on the provided tables. It entails examining the data across each table to find one where the outputs increase at an exponential rate as the inputs increase.
Answer
The function demonstrating exponential growth is obtained from the table with constant output ratios.
Answer for screen readers
The function that demonstrates exponential growth is identified from the table where the ratios of outputs are constant.
Steps to Solve
- Examine the Given Tables
Look carefully at the data in each table provided. Make a note of the inputs (often x-values) and their corresponding outputs (y-values).
- Calculate Ratios Between Successive Outputs
For each table, calculate the ratio of each output to its preceding output. This means for an output $y_n$ and its previous output $y_{n-1}$, compute the ratio:
$$ \text{Ratio} = \frac{y_n}{y_{n-1}} $$
- Check Constant Ratios for Exponential Growth
For a function to be exponential, the ratios calculated in the previous step should be approximately constant. If they vary significantly, the function is not exponential.
- Identify the Function
Once you verify which table has constant ratios, you can conclude that the function from that table demonstrates exponential growth. You can express the function in the form:
$$ f(x) = a \cdot b^x $$
where $a$ is the initial value and $b$ is the base of the exponential growth.
The function that demonstrates exponential growth is identified from the table where the ratios of outputs are constant.
More Information
Exponential functions are commonly represented in the form $f(x) = a \cdot b^x$, where $b > 1$ indicates growth. In exponential growth, the rate of increase is proportional to the current value, which leads to rapid increases.
Tips
- Not calculating ratios correctly; ensure each output is compared properly with its predecessor.
- Misidentifying linear growth as exponential growth due to similar appearances in small data ranges.
- Confusing exponential growth with other types of growth such as logarithmic or polynomial.
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