Answer:
[tex]\text{c)} \quad y=5(2)^x[/tex]
[tex]\text{d)} \quad y=6(2)^x[/tex]
[tex]\text{e)} \quad y=3(7)^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 (c)
Given points:
Substitute the given (x, y) points into the exponential function formula to create two equations:
[tex]\implies 20=ab^2[/tex]
[tex]\implies 640=ab^7[/tex]
Divide the second equation by the first equation to eliminate a:
[tex]\implies \dfrac{640}{20}=\dfrac{ab^7}{ab^2}[/tex]
[tex]\implies 32=\dfrac{b^7}{b^2}[/tex]
Solve for b:
[tex]\implies 32=b^7 \cdot b^{-2}[/tex]
[tex]\implies 32=b^{7-2}[/tex]
[tex]\implies 32=b^5[/tex]
[tex]\implies b=2[/tex]
Substitute the found value of b into one of the equations and solve for a:
[tex]\implies 20=a \cdot 2^2[/tex]
[tex]\implies 20=4a[/tex]
[tex]\implies a=5[/tex]
Substitute the found values of a and b into the exponential function formula:
[tex]\implies y=5(2)^x[/tex]
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Part (d)
Given points:
As a is the y-intercept, a = 6.
Substitute the point (3, 48) and the found value of a into the exponential function formula and solve for b:
[tex]\implies 48=6b^3[/tex]
[tex]\implies 8=b^3[/tex]
[tex]\implies b=2[/tex]
Substitute the found values of a and b into the exponential function formula:
[tex]\implies y=6(2)^x[/tex]
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Part (e)
Given points:
Substitute the given (x, y) points into the exponential function formula to create two equations:
[tex]\implies 21=ab^1[/tex]
[tex]\implies 147=ab^2[/tex]
Divide the second equation by the first equation to eliminate a:
[tex]\implies \dfrac{147}{21}=\dfrac{ab^2}{ab^1}[/tex]
[tex]\implies 7=\dfrac{b^2}{b^1}[/tex]
Solve for b:
[tex]\implies 7=b^2 \cdot b^{-1}[/tex]
[tex]\implies 7=b^{2-1}[/tex]
[tex]\implies 7=b^1[/tex]
[tex]\implies b=7[/tex]
Substitute the found value of b into one of the equations and solve for a:
[tex]\implies 21=7a[/tex]
[tex]\implies a=3[/tex]
Substitute the found values of a and b into the exponential function formula:
[tex]\implies y=3(7)^x[/tex]
