To analyze the data provided and determine the type of function that best fits the dataset, we first look at the \( x \) and \( y \) values given in the table:
- \( x = 0, y = 5 \)
- \( x = 1, y = 35 \)
- \( x = 2, y = 245 \)
- \( x = 3, y = 1715 \)
1. Observing the Values:
Let's examine the \( y \)-values in terms of their changes as \( x \) increases:
- From \( x=0 \) to \( x=1 \), \( y \) changes from \( 5 \) to \( 35 \) (increase by \( 30 \))
- From \( x=1 \) to \( x=2 \), \( y \) changes from \( 35 \) to \( 245 \) (increase by \( 210 \))
- From \( x=2 \) to \( x=3 \), \( y \) changes from \( 245 \) to \( 1715 \) (increase by \( 1470 \))
The increments (30, 210, 1470) are not constant, thus eliminating the possibility of a linear function.
2. Checking for an Exponential Function:
To see if the function could be exponential (of the form \( y = ab^x \)), we can check if the ratio of consecutive \( y \)-values remains constant:
- Ratio for \( y[1]/y[0] = 35/5 = 7 \)
- Ratio for \( y[2]/y[1] = 245/35 = 7 \)
- Ratio for \( y[3]/y[2] = 1715/245 = 7 \)
Since the ratio \( 7 \) remains constant, this indicates an exponential relationship of \( y = 5 \cdot 7^x \).
3. Conclusion:
The data illustrates an exponential function model, specifically:
\[ f(x) = 5 \cdot 7^x \]
So, to answer your question:
- This does not illustrate a linear function or a quadratic function, but it does indeed illustrate an exponential function. The proposed exponential function based on the data matches \( f(x) = 5 \cdot 7^x \).