Answer :
To find an exponential equation that fits the given data, we are looking for an equation of the form [tex]\( y = a \cdot b^x \)[/tex].
Here are the steps to find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] that best fit the data:
1. Data Representation: We have the sets of data [tex]\( x = [1, 2, 3, 4, 5, 6] \)[/tex] and [tex]\( y = [913, 1193, 1591, 2079, 2692, 3553] \)[/tex].
2. Logarithmic Transformation: Since we want to find an exponential relationship, we first transform the [tex]\( y \)[/tex] values using the natural logarithm. This transformation changes the exponential relationship into a linear one, making it easier to find the parameters:
[tex]\[
\ln(y) = \ln(a \cdot b^x) = \ln(a) + x \cdot \ln(b)
\][/tex]
This equation is of the form [tex]\( Y = C + M \cdot x \)[/tex], where [tex]\( Y = \ln(y) \)[/tex], [tex]\( C = \ln(a) \)[/tex], and [tex]\( M = \ln(b) \)[/tex]. Now, it looks like a linear equation.
3. Linear Regression: We perform linear regression on [tex]\( x \)[/tex] versus [tex]\( \ln(y) \)[/tex] to find the best fitting line. This will give us the slope [tex]\( M \)[/tex] and the intercept [tex]\( C \)[/tex].
4. Calculating Parameters:
- The intercept [tex]\( C \)[/tex] from the regression corresponds to [tex]\( \ln(a) \)[/tex]. So to find [tex]\( a \)[/tex], we take the exponential of this intercept: [tex]\( a = e^C \)[/tex].
- The slope [tex]\( M \)[/tex] corresponds to [tex]\( \ln(b) \)[/tex]. So to find [tex]\( b \)[/tex], we take the exponential of this slope: [tex]\( b = e^M \)[/tex].
5. Result: After performing these calculations, we find that the best-fit exponential equation is approximately:
[tex]\[
a \approx 697.440
\][/tex]
[tex]\[
b \approx 1.312
\][/tex]
So, the exponential equation that models the data is [tex]\( y = 697.440 \cdot (1.312)^x \)[/tex].
Here are the steps to find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] that best fit the data:
1. Data Representation: We have the sets of data [tex]\( x = [1, 2, 3, 4, 5, 6] \)[/tex] and [tex]\( y = [913, 1193, 1591, 2079, 2692, 3553] \)[/tex].
2. Logarithmic Transformation: Since we want to find an exponential relationship, we first transform the [tex]\( y \)[/tex] values using the natural logarithm. This transformation changes the exponential relationship into a linear one, making it easier to find the parameters:
[tex]\[
\ln(y) = \ln(a \cdot b^x) = \ln(a) + x \cdot \ln(b)
\][/tex]
This equation is of the form [tex]\( Y = C + M \cdot x \)[/tex], where [tex]\( Y = \ln(y) \)[/tex], [tex]\( C = \ln(a) \)[/tex], and [tex]\( M = \ln(b) \)[/tex]. Now, it looks like a linear equation.
3. Linear Regression: We perform linear regression on [tex]\( x \)[/tex] versus [tex]\( \ln(y) \)[/tex] to find the best fitting line. This will give us the slope [tex]\( M \)[/tex] and the intercept [tex]\( C \)[/tex].
4. Calculating Parameters:
- The intercept [tex]\( C \)[/tex] from the regression corresponds to [tex]\( \ln(a) \)[/tex]. So to find [tex]\( a \)[/tex], we take the exponential of this intercept: [tex]\( a = e^C \)[/tex].
- The slope [tex]\( M \)[/tex] corresponds to [tex]\( \ln(b) \)[/tex]. So to find [tex]\( b \)[/tex], we take the exponential of this slope: [tex]\( b = e^M \)[/tex].
5. Result: After performing these calculations, we find that the best-fit exponential equation is approximately:
[tex]\[
a \approx 697.440
\][/tex]
[tex]\[
b \approx 1.312
\][/tex]
So, the exponential equation that models the data is [tex]\( y = 697.440 \cdot (1.312)^x \)[/tex].
