To find the multiplicative rate of change (also known as the common ratio) of the exponential function given in the table, we need to determine how the value of \( y \) changes as \( x \) increases.
The given points are:
\( (1, \frac{3}{2}) \)
\( (2, \frac{9}{8}) \)
\( (3, \frac{27}{32}) \)
\( (4, \frac{81}{128}) \)
Let's find the common ratio \( r \) by dividing successive \( y \)-values:
1. From \( x = 1 \) to \( x = 2 \):
\[ \frac{\frac{9}{8}}{\frac{3}{2}} = \frac{9}{8} \times \frac{2}{3} = \frac{9 \times 2}{8 \times 3} = \frac{18}{24} = \frac{3}{4} \]
2. From \( x = 2 \) to \( x = 3 \):
\[ \frac{\frac{27}{32}}{\frac{9}{8}} = \frac{27}{32} \times \frac{8}{9} = \frac{27 \times 8}{32 \times 9} = \frac{216}{288} = \frac{3}{4} \]
3. From \( x = 3 \) to \( x = 4 \):
\[ \frac{\frac{81}{128}}{\frac{27}{32}} = \frac{81}{128} \times \frac{32}{27} = \frac{81 \times 32}{128 \times 27} = \frac{2592}{3456} = \frac{3}{4} \]
We can see that the common ratio \( r \) is \( \frac{3}{4} \) in all cases.
Thus, the multiplicative rate of change of the function is **three-fourths**.
The table represents an exponential function.
A 2-column table has 4 rows. The first column is labeled x with entries 1, 2, 3, 4. The second column is labeled y with entries three-halves, nine-eigths, StartFraction 27 Over 32 EndFraction, StartFraction 81 Over 128 EndFraction.
What is the multiplicative rate of change of the function?
Two-thirds
Three-fourths
Four-thirds
Three-halves
1 answer