Question

NO LINKS!!

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:________________________

Answer 1

`\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`

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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`