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Find the exponential function through the following points.

\(\textbf{1)}\) Find the exponential function \(f(x)=a(b)^x\) through \( (0, 5)\) and \((2, 20) \)

Show Function

The answer is \( f(x)=5(2)^{x}\)

Show Work

\(\text{Parent Function}\)

\(\,\,\,\,\,\,f(x)=a\cdot(b)^x\)


\(\text{Plug in the point} (0,5)\)

\(\,\,\,\,\,\,5=a\cdot(b)^0\)

\(\,\,\,\,\,\,5=a\cdot1\)

\(\,\,\,\,\,\,a=5\)


\(\text{Plug in for }a\)

\(\,\,\,\,\,\,f(x)=5\cdot(b)^x\)


\(\text{Plug in the point} (2,20)\)

\(\,\,\,\,\,\,20=5\cdot(b)^2\)

\(\,\,\,\,\,\,4=(b)^2\)

\(\,\,\,\,\,\,\pm2=b\)

\(\,\,\,\,\,\,b=2 \left(\text{note: }b\gt0 \right)\)

The answer is \( f(x)=5(2)^{x}\)

\(\textbf{2)}\) Find the exponential function \(f(x)=a(b)^x\) through \( (0,-60)\) and \((2,-15) \)

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The answer is \( f(x)=-60(.5)^{x}\)

\(\textbf{3)}\) Find the exponential function \(f(x)=a(b)^x\) through \( (0,2)\) and \((3,54) \)

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The answer is \( f(x)=2(3)^{x}\)

\(\textbf{4)}\) Find the exponential function \(f(x)=a(b)^x\) through \( (0,9)\) and \((2,81) \)

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The answer is \( f(x)=9(3)^{x}\)

\(\textbf{5)}\) Find the exponential function \(f(x)=a(b)^x\) through \( (0,16)\) and \((4,1) \)

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The answer is \( f(x)=16(0.5)^{x}\)

\(\textbf{6)}\) Find the exponential function \(f(x)=a(b)^x\) through \( (0,-8)\) and \((3,-1) \)

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The answer is \( f(x)=-8(0.5)^{x}\)

\(\textbf{7)}\) Find the exponential function \(f(x)=a(b)^x\) through \( (0,81)\) and \((4,16) \)

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The answer is \( f(x)=81(2/3)^{x}\)

\(\textbf{8)}\) Find the exponential function \(f(x)=a(b)^x\) through \( (0,8)\) and \((3,27) \)

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The answer is \( f(x)=8(3/2)^{x}\)

\(\textbf{9)}\) Find the exponential function \(f(x)=a(b)^x\) through \( (0,24)\) and \((3,81) \)

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The answer is \( f(x)=24(3/2)^{x}\)

\(\textbf{10)}\) Find the exponential function \(f(x)=a(b)^x\) through \( (0,18)\) and \((2,8) \)

Show Function

The answer is \( f(x)=18(2/3)^{x}\)

See Related Pages\(\)

\(\bullet\text{ Compound Interest}\)
\(\,\,\,\,\,\,\,\,A=P\left(1+\frac{r}{n} \right)^{nt}\,\,\,\,\,\,\,\,A=P e^{rt}\)

\(\bullet\text{ Half Life Questions}\)
\(\,\,\,\,\,\,\,\,A_t=A_0e^{kt}\)

\(\bullet\text{ Graphing Exponential Functions}\)
\(\,\,\,\,\,\,\,\,f(x)=2^{x-1}-2\,\, \)

\(\bullet\text{ Inverse of Exponential Functions}\)
\(\,\,\,\,\,\,\,\,f^{-1}(x)=\log_5(x-1)\)

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