Question

question 1
if all the graphs above have equations with form ( y = ab^x ),
which graph has the largest value for ( b )?
which graph has the smallest value for ( b )?
which graph has the largest value for ( a )?
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question 2
describe the long run behavior of ( f(t) = 3(2)^t + 1 )
as ( t \to -infty, f(t) \to )
as ( t \to infty, f(t) \to )

Explanation:

Step1: Analyze the long - run behavior as \(t\to-\infty\)

For the function \(f(t)=3(2)^{t}+1\), when \(t\to-\infty\), we know that for an exponential function \(y = a\cdot b^{t}\) (\(b> 1\)), as \(t\to-\infty\), \(b^{t}=\frac{1}{b^{-t}}\to0\). Here \(a = 3\) and \(b = 2\). So \(3(2)^{t}\to0\) as \(t\to-\infty\). Then \(f(t)=3(2)^{t}+1\to0 + 1=1\).

Step2: Analyze the long - run behavior as \(t\to\infty\)

For the function \(f(t)=3(2)^{t}+1\), when \(t\to\infty\), since \(b = 2>1\) and \(a=3>0\), the exponential function \(y = 3(2)^{t}\) grows without bound. As \(t\to\infty\), \(3(2)^{t}\to+\infty\). Then \(f(t)=3(2)^{t}+1\to+\infty\)

Answer:

As \(t\to-\infty\), \(f(t)\to1\); As \(t\to\infty\), \(f(t)\to+\infty\)