Question

NO LINKS!! Part 2

Find an exponential function in y = ab^x form that satisfies the given information

c. Passes through the points (2, 20) and (7, 640)

d. passes through (0, 6) and (3, 48)

e. passes through (1, 21) and (2, 147)

Answer 1

`\text{c)} \quad y=5(2)^x`

`\text{d)} \quad y=6(2)^x`

`\text{e)} \quad y=3(7)^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 (c)

Given points:

• (2, 20)
• (7, 640)

Substitute the given (x, y) points into the exponential function formula to create two equations:

`\implies 20=ab^2`

`\implies 640=ab^7`

Divide the second equation by the first equation to eliminate a:

`\implies (640)/(20)=(ab^7)/(ab^2)`

`\implies 32=(b^7)/(b^2)`

Solve for b:

`\implies 32=b^7 \cdot b^(-2)`

`\implies 32=b^(7-2)`

`\implies 32=b^5`

`\implies b=2`

Substitute the found value of b into one of the equations and solve for a:

`\implies 20=a \cdot 2^2`

`\implies 20=4a`

`\implies a=5`

Substitute the found values of a and b into the exponential function formula:

`\implies y=5(2)^x`

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Part (d)

Given points:

• (0, 6)
• (3, 48)

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:

`\implies 48=6b^3`

`\implies 8=b^3`

`\implies b=2`

Substitute the found values of a and b into the exponential function formula:

`\implies y=6(2)^x`

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Part (e)

Given points:

• (1, 21)
• (2, 147)

Substitute the given (x, y) points into the exponential function formula to create two equations:

`\implies 21=ab^1`

`\implies 147=ab^2`

Divide the second equation by the first equation to eliminate a:

`\implies (147)/(21)=(ab^2)/(ab^1)`

`\implies 7=(b^2)/(b^1)`

Solve for b:

`\implies 7=b^2 \cdot b^(-1)`

`\implies 7=b^(2-1)`

`\implies 7=b^1`

`\implies b=7`

Substitute the found value of b into one of the equations and solve for a:

`\implies 21=7a`

`\implies a=3`

Substitute the found values of a and b into the exponential function formula:

`\implies y=3(7)^x`