Question

NO LINKS!!!! URGENT HELP PLEASE!!!!

If f(x) is an exponential function where f(-1) = 30 and f(4) = 85, then find the value of f(3), to the nearest hundredth.

Answer 1

Answer: 69.02

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Work Shown:

The notation f(-1) = 30 tells us that (x,y) = (-1,30) is a point on the exponential curve.

Plug in x = -1 and y = 30 and solve for 'a'

y = a*b^x

30 = a*b^(-1)

30 = a*(1/b^1)

30 = a/b

a = 30b

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Now plug that into the equation to get

y = a*b^x

y = 30b*b^x

y = 30b^1*b^x

y = 30b^(1+x)

Afterward, plug in x = 4 and y = 85 to solve for b.

85 = 30b^(1+4)

85 = 30b^5

b^5 = 85/30

b = (85/30)^(1/5)

b = 1.23157122757057

We can then determine the value of 'a'

a = 30b

a = 30*1.23157122757057

a = 36.947136827117

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To summarize,

• a = 36.947136827117
• b = 1.23157122757057

both of which are approximate.

Since we want f(3) to the nearest hundredth, we don't really need to go too overboard with precision. Let's round 'a' and b to 4 decimal places.

We then get

• a = 36.9471
• b = 1.2316

This means

y = a*b^x

turns into

y = 36.9471*(1.2316)^x

As a check, plug in x = -1 to get y = 29.999269; we'll have rounding error since we rounded those 'a' and b values earlier. But when rounding 29.999269 to the nearest hundredth, we get y = 30.

Also, plug x = 4 into the equation to get y = 85.00785875 which rounds to 85.

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The last step is to plug x = 3 into the equation

y = 36.9471*(1.2316)^x

y = 36.9471*(1.2316)^3

y = 69.0222951885528

y = 69.02 which is the final answer

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If you were to use those more accurate values of 'a' and b, then x = 3 would lead to y = 69.0175266335801; that rounds to 69.02 when rounding to the nearest hundredth (aka 2 decimal places). So this shows we can stick with the less accurate 'a' and b values while still getting the correct final answer.

GeoGebra and Desmos are great tools to help quickly verify the answer.

Answer 2

f(3) = 69.02 (nearest hundredth)

Explanation:

Exponential Function

An exponential function is used to calculate the exponential growth or decay of a given set of data. In an exponential function, the variable is the exponent.

`\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}}`

Given:

• f(-1) = 30
• f(4) = 85

Substitute these values into the general form of an exponential function to create two equations in terms of a and b:

`\begin{aligned}\implies f(-1)&=30\\ab^(-1)&=30\end{aligned}`

`\begin{aligned}\implies f(4)&=85\\ab^(4)&=85\end{aligned}`

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

`\begin{aligned}\implies (ab^4)/(ab^(-1))&=(85)/(30)\\\\(b^4)/(b^(-1))&=(85)/(30)\\\\\end{aligned}`

Solve for b:

`\begin{aligned}\implies (b^4)/(b^(-1))&=(85)/(30)\\\\b^(4-(-1))&=(17)/(6)\\\\b^(5)&=(17)/(6)\\\\b&=\sqrt[5]{(17)/(6)}\end{aligned}`

Substitute the found value of b into ab⁴ = 85:

`\implies a\left(\sqrt[5]{(17)/(6)}\right)^4=85`

Solve for a:

`\implies a=\frac{85}{\left(\sqrt[5]{(17)/(6)}\right)^4}`

`\implies a=\frac{85}{\sqrt[5]{(17^4)/(6^4)}}`

`\implies a=\frac{85}{\sqrt[5]{(83521)/(1296)}}`

Substitute the found values of a and b into the exponential function formula to create an exponential equation for the given parameters:

`f(x)=\left(\frac{85}{\sqrt[5]{(83521)/(1296)}}\right)\cdot \left(\sqrt[5]{(17)/(6)}\right)^x`

To find the value of f(3), substitute x = 3 into the equation:

`\begin{aligned}\implies f(3)&=\left(\frac{85}{\sqrt[5]{(83521)/(1296)}}\right)\cdot \left(\sqrt[5]{(17)/(6)}\right)^3\\\\&=69.0175266...\\\\&=69.02\; \;\sf (nearest\;hundredth)\end{aligned}`

Therefore, the value of f(3) to the nearest hundredth is 69.02.