Sure, let's analyze the data points given in the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
4 & 128 \\
\hline
5 & 200 \\
\hline
6 & 288 \\
\hline
7 & 392 \\
\hline
8 & 512 \\
\hline
\end{array}
\][/tex]
Based on the problem, we need to find a function that accurately models this data. We consider the three types:
1. Linear [tex]\( y = mx + b \)[/tex]
2. Quadratic [tex]\( y = ax^2 \)[/tex]
3. Exponential [tex]\( y = a(b)^x \)[/tex]
Let’s systematically evaluate each type:
1. Linear Function [tex]\( y = mx + b \)[/tex]:
- In a linear relationship, the change in [tex]\( y \)[/tex] should be proportional to the change in [tex]\( x \)[/tex]. We notice smaller rate of differences between successive [tex]\( y \)[/tex] values as [tex]\( x \)[/tex] increases. Thus, linear is unlikely to be suitable for this data.
2. Exponential Function [tex]\( y = a(b)^x \)[/tex]:
- For an exponential function, the ratio [tex]\( \frac{y_{n+1}}{y_n} \)[/tex] should remain roughly constant. Observing the data:
[tex]\[
\frac{200}{128} \approx 1.5625, \quad \frac{288}{200} \approx 1.44, \quad \frac{392}{288} \approx 1.3611, \quad \frac{512}{392} \approx 1.3061
\][/tex]
The ratios are not consistent, suggesting that an exponential model does not fit well.
3. Quadratic Function [tex]\( y = ax^2 \)[/tex]:
- For a quadratic relationship, we expect the values of [tex]\( \frac{y}{x^2} \)[/tex] to be constant:
[tex]\[
\frac{128}{4^2} = \frac{128}{16} = 8, \quad \frac{200}{5^2} = \frac{200}{25} = 8, \quad \frac{288}{6^2} = \frac{288}{36} = 8, \quad \frac{392}{7^2} = \frac{392}{49} = 8, \quad \frac{512}{8^2} = \frac{512}{64} = 8
\][/tex]
Since all the ratios are consistently [tex]\( 8 \)[/tex], we conclude that the best fitting function is a quadratic function of the form:
[tex]\[ y = 8x^2 \][/tex]
Therefore, the function that models the given data is:
[tex]\[ y = 8x^2 \][/tex]
