Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Experience the ease of finding accurate answers to your questions from a knowledgeable community of professionals. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
To determine the multiplicative rate of change of the given exponential function \( y = a \cdot r^x \), we look at the ratios of consecutive \( y \) values for given \( x \) values. This is because, in an exponential function, each term is the previous term multiplied by a constant rate, \( r \).
Given:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & \frac{3}{2} \\ \hline 2 & \frac{9}{8} \\ \hline 3 & \frac{27}{32} \\ \hline 4 & \frac{81}{128} \\ \hline \end{array} \][/tex]
To find \( r \), we calculate the ratio of \( y \) values using the following steps:
1. Calculate \( r \) from \( x = 1 \) and \( x = 2 \):
[tex]\[ r = \frac{y(2)}{y(1)} = \frac{\frac{9}{8}}{\frac{3}{2}} = \frac{9}{8} \times \frac{2}{3} = \frac{9 \cdot 2}{8 \cdot 3} = \frac{18}{24} = \frac{3}{4} \][/tex]
2. Calculate \( r \) from \( x = 2 \) and \( x = 3 \):
[tex]\[ r = \frac{y(3)}{y(2)} = \frac{\frac{27}{32}}{\frac{9}{8}} = \frac{27}{32} \times \frac{8}{9} = \frac{27 \cdot 8}{32 \cdot 9} = \frac{216}{288} = \frac{3}{4} \][/tex]
3. Calculate \( r \) from \( x = 3 \) and \( x = 4 \):
[tex]\[ r = \frac{y(4)}{y(3)} = \frac{\frac{81}{128}}{\frac{27}{32}} = \frac{81}{128} \times \frac{32}{27} = \frac{81 \cdot 32}{128 \cdot 27} = \frac{2592}{3456} = \frac{3}{4} \][/tex]
In each case, we find that the ratio \( r \) is constant and equals \( \frac{3}{4} \).
Thus, the multiplicative rate of change of the function is \( \frac{3}{4} \).
The correct answer is:
[tex]\(\boxed{\frac{3}{4}}\)[/tex]
Given:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & \frac{3}{2} \\ \hline 2 & \frac{9}{8} \\ \hline 3 & \frac{27}{32} \\ \hline 4 & \frac{81}{128} \\ \hline \end{array} \][/tex]
To find \( r \), we calculate the ratio of \( y \) values using the following steps:
1. Calculate \( r \) from \( x = 1 \) and \( x = 2 \):
[tex]\[ r = \frac{y(2)}{y(1)} = \frac{\frac{9}{8}}{\frac{3}{2}} = \frac{9}{8} \times \frac{2}{3} = \frac{9 \cdot 2}{8 \cdot 3} = \frac{18}{24} = \frac{3}{4} \][/tex]
2. Calculate \( r \) from \( x = 2 \) and \( x = 3 \):
[tex]\[ r = \frac{y(3)}{y(2)} = \frac{\frac{27}{32}}{\frac{9}{8}} = \frac{27}{32} \times \frac{8}{9} = \frac{27 \cdot 8}{32 \cdot 9} = \frac{216}{288} = \frac{3}{4} \][/tex]
3. Calculate \( r \) from \( x = 3 \) and \( x = 4 \):
[tex]\[ r = \frac{y(4)}{y(3)} = \frac{\frac{81}{128}}{\frac{27}{32}} = \frac{81}{128} \times \frac{32}{27} = \frac{81 \cdot 32}{128 \cdot 27} = \frac{2592}{3456} = \frac{3}{4} \][/tex]
In each case, we find that the ratio \( r \) is constant and equals \( \frac{3}{4} \).
Thus, the multiplicative rate of change of the function is \( \frac{3}{4} \).
The correct answer is:
[tex]\(\boxed{\frac{3}{4}}\)[/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.