Answer:
[tex]\text{f)} \quad y=80(1.1)^x[/tex]
[tex]\text{g)} \quad y=225(0.8)^x[/tex]
[tex]\text{h)} \quad y=5(3)^x[/tex]
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function}\\\\$y=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}[/tex]
Part (f)
Given points:
Substitute the given (x, y) points into the exponential function formula to create two equations:
[tex]\implies 72.73=ab^{-1}[/tex]
[tex]\implies 106.48=ab^3[/tex]
Divide the second equation by the first equation to eliminate a:
[tex]\implies \dfrac{106.48}{72.73}=\dfrac{ab^3}{ab^{-1}}[/tex]
[tex]\implies \dfrac{106.48}{72.73}=\dfrac{b^3}{b^{-1}}[/tex]
Solve for b:
[tex]\implies \dfrac{106.48}{72.73}=b^3 \cdot b^{1}[/tex]
[tex]\implies \dfrac{106.48}{72.73}=b^{3+1}[/tex]
[tex]\implies \dfrac{106.48}{72.73}=b^4[/tex]
[tex]\implies b=1.09998968...[/tex]
[tex]\implies b=1.1[/tex]
Substitute the found value of b into one of the equations and solve for a:
[tex]\implies 72.73=a \cdot (1.09998968...)^{-1}[/tex]
[tex]\implies a=80.0022499...[/tex]
[tex]\implies a=80[/tex]
Substitute the found values of a and b into the exponential function formula:
[tex]\implies y=80(1.1)^x[/tex]
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Part (g)
Given points:
- (-2, 351.56225)
- (3, 115.2)
Substitute the given (x, y) points into the exponential function formula to create two equations:
[tex]\implies 351.56225=ab^{-2}[/tex]
[tex]\implies 115.2=ab^3[/tex]
Divide the second equation by the first equation to eliminate a:
[tex]\implies \dfrac{115.2}{351.56225}=\dfrac{ab^3}{ab^{-2}}[/tex]
[tex]\implies \dfrac{115.2}{351.56225}=\dfrac{b^3}{b^{-2}}[/tex]
Solve for b:
[tex]\implies\dfrac{115.2}{351.56225}=b^3 \cdot b^{2}[/tex]
[tex]\implies \dfrac{115.2}{351.56225}=b^{3+2}[/tex]
[tex]\implies \dfrac{115.2}{351.56225}=b^5[/tex]
[tex]\implies b=0.800000113...[/tex]
[tex]\implies b=0.8[/tex]
Substitute the found value of b into one of the equations and solve for a:
[tex]\implies 115.2=a(0.800000113...)^3[/tex]
[tex]\implies a=224.999904[/tex]
[tex]\implies a=225[/tex]
Substitute the found values of a and b into the exponential function formula:
[tex]\implies y=225(0.8)^x[/tex]
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Part (h)
Given points:
Substitute the given (x, y) points into the exponential function formula to create two equations:
[tex]\implies 405=ab^4[/tex]
[tex]\implies 98415=ab^9[/tex]
Divide the second equation by the first equation to eliminate a:
[tex]\implies \dfrac{98415}{405}=\dfrac{ab^9}{ab^4}[/tex]
[tex]\implies 243=\dfrac{b^9}{b^4}[/tex]
Solve for b:
[tex]\implies 243=b^9 \cdot b^{-4}[/tex]
[tex]\implies 243=b^{9-4}[/tex]
[tex]\implies 243=b^{5}[/tex]
[tex]\implies b=3[/tex]
Substitute the found value of b into one of the equations and solve for a:
[tex]\implies 405=a \cdot 3^4[/tex]
[tex]\implies 81a=405[/tex]
[tex]\implies a=5[/tex]
Substitute the found values of a and b into the exponential function formula:
[tex]\implies y=5(3)^x[/tex]
