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Write an equation to represent the following graphs:
a. passes through (-2, 18.75 and (1, 1, 1.2)
a-value:___________ b-value:____________
Equation: _____________________

b. passes through (0, 6) and (2, 8.64)
a-value:_____________ b-value:_____________
Equation:________________________

User Prossellob
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1 Answer

5 votes

Answer:


\textsf{a)\;\;passes\;through\;$(-2, 18.75)$\;and\;$(1, 1.2)$}


\textsf{$a$-value:\;\;3 \quad $b$-value: \;\;0.4}


\textsf{Equation: \quad $y=3 (0.4)^x$}


\textsf{b)\;\;passes\;through\;$(0, 6)$\;and\;$(2, 8.64)$}


\textsf{$a$-value:\;\;6 \quad $b$-value: \;\;1.2}


\textsf{Equation: \quad $y=6 (1.2)^x$}

Explanation:


\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}}

Part (a)

Given points:

  • (-2, 18.75)
  • (1, 1.2)

Substitute both points into the exponential function formula to create two equations:


  • \textsf{Equation\;1}: \quad 18.75=ab^(-2)

  • \textsf{Equation\;2}: \quad 1.2=ab

Divide the equations to eliminate a, then solve for b:


\implies (18.75)/(1.2)=(ab^(-2))/(ab)


\implies15.625=(b^(-2))/(b)


\implies15.625=b^(-2)b^(-1)


\implies15.625=b^(-3)


\implies15.625=(1)/(b^(3))


\implies b^(3)=(1)/(15.625)


\implies b=0.4

Substitute the found value of b into the second equation and solve for b:


\implies 1.2=0.4a


\implies a=3

Therefore, the exponential equation is:


y=3 (0.4)^x

-------------------------------------------------------------------------------------------

Part (b)

Given points:

  • (0, 6)
  • (2, 8.64)

Substitute point (0, 6) into the exponential function formula and solve for a:


\implies 6=ab^0


\implies 6=a(1)


\implies a=6

Substitute the found value of a and point (2, 8.64) into the exponential function formula and solve for b:


\implies 8.64=6b^2


\implies b^2=(8.64)/(6)


\implies b^2=1.44


\implies b=√(1.44)


\implies b=1.2

Therefore, the exponential equation is:


y=6 (1.2)^x

User Mahabub
by
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