To model the given data, we need to determine whether the data fits into a linear, quadratic, or exponential function. Let's analyze the given data points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 15 \\
\hline
2 & 45 \\
\hline
3 & 135 \\
\hline
4 & 405 \\
\hline
5 & 1215 \\
\hline
\end{array}
\][/tex]
Let's consider each function type step-by-step:
1. Linear Function [tex]\(y = mx + b\)[/tex]:
- If the data followed a linear function, the difference between consecutive [tex]\(y\)[/tex]-values would be constant.
- Calculate the differences: [tex]\(45 - 15 = 30\)[/tex], [tex]\(135 - 45 = 90\)[/tex], [tex]\(405 - 135 = 270\)[/tex], [tex]\(1215 - 405 = 810\)[/tex].
- The differences are not constant, so the data does not fit a linear function.
2. Quadratic Function [tex]\(y = ax^2\)[/tex]:
- If the data followed a quadratic function, the second differences (differences of differences) would be constant.
- First differences: [tex]\(45 - 15 = 30\)[/tex], [tex]\(135 - 45 = 90\)[/tex], [tex]\(405 - 135 = 270\)[/tex], [tex]\(1215 - 405 = 810\)[/tex].
- Second differences: [tex]\(90 - 30 = 60\)[/tex], [tex]\(270 - 90 = 180\)[/tex], [tex]\(810 - 270 = 540\)[/tex].
- The second differences are not constant, so the data does not fit a quadratic function.
3. Exponential Function [tex]\(y = a(b)^x\)[/tex]:
- If the data followed an exponential function, the ratio of consecutive [tex]\(y\)[/tex]-values would be constant.
- Calculate the ratios: [tex]\(\frac{45}{15} = 3.0\)[/tex], [tex]\(\frac{135}{45} = 3.0\)[/tex], [tex]\(\frac{405}{135} = 3.0\)[/tex], [tex]\(\frac{1215}{405} = 3.0\)[/tex].
- The ratios are all constant and equal to [tex]\(3.0\)[/tex].
Given that the data points fit an exponential function, we'll determine the parameters [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the function [tex]\(y = a(b)^x\)[/tex].
- From the ratios, we see that [tex]\(b = 3.0\)[/tex].
- To find [tex]\(a\)[/tex], we use the first data point [tex]\((x = 1, y = 15)\)[/tex]:
[tex]\[
15 = a \cdot (3.0)^1 \implies 15 = 3a \implies a = \frac{15}{3} = 5.0
\][/tex]
Therefore, the exponential function that models the data is:
[tex]\[
\boxed{y = 5.0 \times (3.0)^x}
\][/tex]
