Answer :
To determine which set of ordered pairs could be generated by an exponential function, let's analyze each set individually for the property of exponential growth.
An exponential function can be generally expressed as [tex]\( y = ab^x \)[/tex], where [tex]\( a \)[/tex] is a constant and [tex]\( b \)[/tex] is the base of the exponential. The key characteristic of an exponential function is that the ratio [tex]\( \frac{y_{i+1}}{y_i} \)[/tex] between successive [tex]\( y \)[/tex]-values should be constant.
Set 1: [tex]\((0,0), (1,1), (2,8), (3,27)\)[/tex]
- [tex]\( \frac{y_2}{y_1} = \frac{1}{0} \)[/tex] (undefined, division by zero)
Thus, this set cannot be exponential since it begins with zero, which makes the ratios undefined.
Set 2: [tex]\((0,1), (1,2), (2,5), (3,10)\)[/tex]
Calculate the ratios:
- [tex]\( \frac{y_2}{y_1} = \frac{2}{1} = 2 \)[/tex]
- [tex]\( \frac{y_3}{y_2} = \frac{5}{2} = 2.5 \)[/tex]
- [tex]\( \frac{y_4}{y_3} = \frac{10}{5} = 2 \)[/tex]
The ratios are not consistent, so this set cannot be generated by an exponential function.
Set 3: [tex]\((0,0), (1,3), (2,6), (3,9)\)[/tex]
- [tex]\( \frac{y_2}{y_1} = \frac{3}{0} \)[/tex] (undefined, division by zero)
Again, division by zero indicates this set cannot exhibit exponential behavior.
Set 4: [tex]\((0,1), (1,3), (2,9), (3,27)\)[/tex]
Calculate the ratios:
- [tex]\( \frac{y_2}{y_1} = \frac{3}{1} = 3 \)[/tex]
- [tex]\( \frac{y_3}{y_2} = \frac{9}{3} = 3 \)[/tex]
- [tex]\( \frac{y_4}{y_3} = \frac{27}{9} = 3 \)[/tex]
The ratios are all 3, showing a consistent multiplication factor. Therefore, the ordered pairs in Set 4 can be generated by an exponential function.
Conclusion: The set of ordered pairs that could be generated by an exponential function is [tex]\((0,1), (1,3), (2,9), (3,27)\)[/tex].
An exponential function can be generally expressed as [tex]\( y = ab^x \)[/tex], where [tex]\( a \)[/tex] is a constant and [tex]\( b \)[/tex] is the base of the exponential. The key characteristic of an exponential function is that the ratio [tex]\( \frac{y_{i+1}}{y_i} \)[/tex] between successive [tex]\( y \)[/tex]-values should be constant.
Set 1: [tex]\((0,0), (1,1), (2,8), (3,27)\)[/tex]
- [tex]\( \frac{y_2}{y_1} = \frac{1}{0} \)[/tex] (undefined, division by zero)
Thus, this set cannot be exponential since it begins with zero, which makes the ratios undefined.
Set 2: [tex]\((0,1), (1,2), (2,5), (3,10)\)[/tex]
Calculate the ratios:
- [tex]\( \frac{y_2}{y_1} = \frac{2}{1} = 2 \)[/tex]
- [tex]\( \frac{y_3}{y_2} = \frac{5}{2} = 2.5 \)[/tex]
- [tex]\( \frac{y_4}{y_3} = \frac{10}{5} = 2 \)[/tex]
The ratios are not consistent, so this set cannot be generated by an exponential function.
Set 3: [tex]\((0,0), (1,3), (2,6), (3,9)\)[/tex]
- [tex]\( \frac{y_2}{y_1} = \frac{3}{0} \)[/tex] (undefined, division by zero)
Again, division by zero indicates this set cannot exhibit exponential behavior.
Set 4: [tex]\((0,1), (1,3), (2,9), (3,27)\)[/tex]
Calculate the ratios:
- [tex]\( \frac{y_2}{y_1} = \frac{3}{1} = 3 \)[/tex]
- [tex]\( \frac{y_3}{y_2} = \frac{9}{3} = 3 \)[/tex]
- [tex]\( \frac{y_4}{y_3} = \frac{27}{9} = 3 \)[/tex]
The ratios are all 3, showing a consistent multiplication factor. Therefore, the ordered pairs in Set 4 can be generated by an exponential function.
Conclusion: The set of ordered pairs that could be generated by an exponential function is [tex]\((0,1), (1,3), (2,9), (3,27)\)[/tex].
