Answer:
[tex]y=3^x+10[/tex]
f(x)-values
decreased by 4
(4,87)
Step-by-step explanation:
The table shows the exponential growth function
[tex]y=a\cdot b^x +c[/tex]
substitute some values to find [tex]a,b,c:[/tex]
[tex]13=a\cdot b^1+c\\ \\19=a\cdot b^2 +c\\ \\37=a\cdot b^3+c[/tex]
Subtract these equations:
[tex]ab^2+c-ab-c=19-13\Rightarrow ab(b-1)=6\\ \\ab^3+c-ab^2-c=37-19\Rightarrow ab^2(b-1)=18[/tex]
Divide them:
[tex]\dfrac{ab^2(b-1)}{ab(b-1)}=\dfrac{18}{6}\Rightarrow b=3[/tex]
Then
[tex]3a(3-1)=6\Rightarrow a=1[/tex]
Hence,
[tex]13=1\cdot 3+c\Rightarrow c=10[/tex]
Therefore, the parent function is [tex]y=3^x+10[/tex]
If this function would be translated 4 units down, its expression will be
[tex]y=3^x+10-4\\ \\y=3^x+6[/tex]
This means that f(x)-values decreased by 4.
Then the table for translated function is
[tex]\begin{array}{cccccc}x&1&2&3&4&5\\ \\f(x)&9&15&33&87&249\end{array}[/tex]
The graph of translated function passes through the point (4,87)
