Answer:
x y
-2 7
-1 10.5
0 15.75
1 23.625
2 35.4375
Step-by-step explanation:
The general equation of the exponential function is [tex]y=ab^x[/tex].
We know from our table that when [tex]x=0[/tex], [tex]y=15.75[/tex]. Let's replace those values in our equation:
[tex]y=ab^x[/tex]
[tex]15.75=ab^0[/tex]
Remember that [tex]b^0=1[/tex], so:
[tex]15.75=a(1)[/tex]
[tex]15.75=a[/tex]
[tex]a=15.75[/tex]
We also know from our table that when [tex]x=-1[/tex], [tex]y=10.5[/tex]. Let's replace the values again:
[tex]y=ab^x[/tex]
[tex]10.5=ab^{-1}[/tex]
But we now know that [tex]a=15.75[/tex], so let's replace that value as well:
[tex]10.5=15.75b^{-1}[/tex]
Remember that [tex]b^{-1}=\frac{1}{b}[/tex], so:
[tex]10.5=\frac{15.75}{b}[/tex]
[tex]10.5b=15.75[/tex]
[tex]b=\frac{15.75}{10.5}[/tex]
[tex]b=1.5[/tex]
Now, we can put it all together to complete our exponential function:
[tex]y=ab^x[/tex]
[tex]y=15.75(1.5)^x[/tex]
To find the missing values, we just need to evaluate our function at [tex]x=1[/tex] and [tex]x=2[/tex]:
- For [tex]x=1[/tex]
[tex]y=15.75(1.5)^x[/tex]
[tex]y=15.75(1.5)^1[/tex]
[tex]y=23.625[/tex]
- For [tex]x=2[/tex]
[tex]y=15.75(1.5)^x[/tex]
[tex]y=15.75(1.5)^2[/tex]
[tex]y=35.4375[/tex]
