To find the missing values for the exponential function represented by the table, we can follow these steps:
1. Identify the known values in the table for the function [tex]\( y \)[/tex]:
- When [tex]\( x = -2 \)[/tex], [tex]\( y = 9 \)[/tex]
- When [tex]\( x = -1 \)[/tex], [tex]\( y = 13.5 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 20.25 \)[/tex]
2. Determine the common ratio [tex]\( r \)[/tex] for the exponential function. The common ratio between consecutive terms in an exponential function is given by:
[tex]\[
r = \frac{y_{next}}{y_{previous}}
\][/tex]
Using the values at [tex]\( x = -1 \)[/tex] and [tex]\( x = -2 \)[/tex]:
[tex]\[
r = \frac{13.5}{9}
\][/tex]
3. Calculate the value of [tex]\( r \)[/tex]:
[tex]\[
r = 1.5
\][/tex]
4. Use the common ratio [tex]\( r \)[/tex] to find the missing values. For [tex]\( x = 1 \)[/tex]:
[tex]\[
y_{1} = y_{0} \cdot r = 20.25 \cdot 1.5 = 30.375
\][/tex]
5. Similarly, for [tex]\( x = 2 \)[/tex]:
[tex]\[
y_{2} = y_{1} \cdot r = 30.375 \cdot 1.5 = 45.5625
\][/tex]
Therefore, the missing values for [tex]\( y \)[/tex] when [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex] are [tex]\( 30.375 \)[/tex] and [tex]\( 45.5625 \)[/tex] respectively.
Based on the provided choices:
a. [tex]\( 13.5 \)[/tex]
b. [tex]\(-30.375\)[/tex]
c. [tex]\(30.375\)[/tex]
d. [tex]\(-30.375\)[/tex] and [tex]\( -45.5625 \)[/tex]
e. [tex]\(45.5625\)[/tex]
The best answer is:
c. [tex]\( 30.375 \)[/tex]
45.5625
