Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Discover a wealth of knowledge from professionals across various disciplines on our user-friendly Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To determine which table represents an exponential function of the form [tex]\( y = b^x \)[/tex] with [tex]\( 0 < b < 1 \)[/tex], let's analyze each table carefully.
### First Table Analysis:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & \frac{1}{27} \\ \hline -2 & \frac{1}{9} \\ \hline -2 & \frac{1}{3} \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & 9 \\ \hline 3 & 27 \\ \hline \end{array} \][/tex]
- For [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]: [tex]\( \frac{1/9}{1/27} = 3 \)[/tex]
- For [tex]\( x = -2 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\( \frac{1}{1/9} = 9 \)[/tex]
- For [tex]\( x = 0 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( \frac{9}{1} = 9 \)[/tex]
- For [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]: [tex]\( \frac{27}{9} = 3 \)[/tex]
We can see that the values don't maintain a consistent ratio, suggesting it is not an exponential function of the desired form.
### Second Table Analysis:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & 27 \\ \hline -2 & 9 \\ \hline -1 & 3 \\ \hline 0 & 1 \\ \hline 1 & \frac{1}{3} \\ \hline 2 & \frac{1}{9} \\ \hline 3 & \frac{1}{27} \\ \hline \end{array} \][/tex]
- For [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]: [tex]\( \frac{9}{27} = \frac{1}{3} \)[/tex]
- For [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex]: [tex]\( \frac{3}{9} = \frac{1}{3} \)[/tex]
- For [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\( \frac{1}{3} = \frac{1}{3} \)[/tex]
- For [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\( \frac{1/3}{1} = \frac{1}{3} \)[/tex]
- For [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( \frac{1/9}{1/3} = \frac{1}{3} \)[/tex]
- For [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]: [tex]\( \frac{1/27}{1/9} = \frac{1}{3} \)[/tex]
Each ratio [tex]\( \frac{y_{n+1}}{y_{n}} \)[/tex] is [tex]\( \frac{1}{3} \)[/tex], which is consistent and within the range [tex]\( 0 < b < 1 \)[/tex].
Therefore, the second table represents an exponential function of the form [tex]\( y = b^x \)[/tex] with [tex]\( 0 < b < 1 \)[/tex].
The table number is [tex]\( \boxed{2} \)[/tex].
### First Table Analysis:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & \frac{1}{27} \\ \hline -2 & \frac{1}{9} \\ \hline -2 & \frac{1}{3} \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & 9 \\ \hline 3 & 27 \\ \hline \end{array} \][/tex]
- For [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]: [tex]\( \frac{1/9}{1/27} = 3 \)[/tex]
- For [tex]\( x = -2 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\( \frac{1}{1/9} = 9 \)[/tex]
- For [tex]\( x = 0 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( \frac{9}{1} = 9 \)[/tex]
- For [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]: [tex]\( \frac{27}{9} = 3 \)[/tex]
We can see that the values don't maintain a consistent ratio, suggesting it is not an exponential function of the desired form.
### Second Table Analysis:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & 27 \\ \hline -2 & 9 \\ \hline -1 & 3 \\ \hline 0 & 1 \\ \hline 1 & \frac{1}{3} \\ \hline 2 & \frac{1}{9} \\ \hline 3 & \frac{1}{27} \\ \hline \end{array} \][/tex]
- For [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]: [tex]\( \frac{9}{27} = \frac{1}{3} \)[/tex]
- For [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex]: [tex]\( \frac{3}{9} = \frac{1}{3} \)[/tex]
- For [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\( \frac{1}{3} = \frac{1}{3} \)[/tex]
- For [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\( \frac{1/3}{1} = \frac{1}{3} \)[/tex]
- For [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( \frac{1/9}{1/3} = \frac{1}{3} \)[/tex]
- For [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]: [tex]\( \frac{1/27}{1/9} = \frac{1}{3} \)[/tex]
Each ratio [tex]\( \frac{y_{n+1}}{y_{n}} \)[/tex] is [tex]\( \frac{1}{3} \)[/tex], which is consistent and within the range [tex]\( 0 < b < 1 \)[/tex].
Therefore, the second table represents an exponential function of the form [tex]\( y = b^x \)[/tex] with [tex]\( 0 < b < 1 \)[/tex].
The table number is [tex]\( \boxed{2} \)[/tex].
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
