Discover how IDNLearn.com can help you find the answers you need quickly and easily. Our platform offers detailed and accurate responses from experts, helping you navigate any topic with confidence.

Determine the equations of each of the exponential functions represented below.

a)

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
7 & 1250 \\
\hline
14 & 500 \\
\hline
21 & 200 \\
\hline
28 & 80 \\
\hline
35 & 32 \\
\hline
\end{tabular}


Sagot :

To determine the equation of an exponential function given a set of data points, we follow a systematic approach. The data is provided as follows:

[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 7 & 1250 \\ \hline 14 & 500 \\ \hline 21 & 200 \\ \hline 28 & 80 \\ \hline 35 & 32 \\ \hline \end{array} \][/tex]

The goal is to find an exponential function of the form [tex]\( y = ab^x \)[/tex].

### Step 1: Linearizing the Exponential Relation

Firstly, we take the natural logarithm of the y-values to transform the exponential relation into a linear form:

[tex]\[ \begin{array}{|c|c|c|} \hline x & y & \log(y) \\ \hline 7 & 1250 & 7.1309 \\ \hline 14 & 500 & 6.2146 \\ \hline 21 & 200 & 5.2983 \\ \hline 28 & 80 & 4.3820 \\ \hline 35 & 32 & 3.4657 \\ \hline \end{array} \][/tex]

### Step 2: Setting Up the Linear Regression

We assume a linear relationship in the form:

[tex]\[ \log(y) = \log(a) + x \cdot \log(b) \][/tex]

Here, the x-values remain the same, and the y-values are now the natural logarithms of the original y-values.

### Step 3: Performing Linear Regression

Using the linearized form [tex]\(\log(y) = m \cdot x + c\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(c\)[/tex] is the y-intercept:

From our calculations, we find:

[tex]\[ m = -0.1309 \][/tex]
[tex]\[ c = 8.0472 \][/tex]

### Step 4: Back-Transforming to Exponential Form

Solving for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:

[tex]\[ \log(b) = m \][/tex]
[tex]\[ \log(a) = c \][/tex]

Thus:

[tex]\[ b = e^{\log(b)} = e^{-0.1309} \approx 0.8773 \][/tex]
[tex]\[ a = e^{\log(a)} = e^{8.0472} \approx 3125 \][/tex]

### Step 5: Writing the Final Equation

With our derived constants [tex]\(a\)[/tex] and [tex]\(b\)[/tex], we can write the exponential function as:

[tex]\[ y = 3125 \cdot (0.8773)^x \][/tex]

This is the exponential function that fits the given data points accurately.