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The growth rate of the function $$f(x) = b a ^ x$$ is $17\%$, and $f(0) = 24$

What I am trying to figure out is how to find out what $a$ and $b$ in this equation are?

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    What happens when you substitute $x=0$ into the given expression for $f(x)$?2012-11-21
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    Do you know some calculus?2012-11-21
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    As it says, f(0) = 24 - all that i know is above2012-11-21
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    Hint 1: f(0) = b a^x = b a^0 = b = 24. Is 17 the value at some given x or for some value for 'a'?2012-11-21
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    17% is the growth of the function, it's expotential, it's raising with 17% for each time it moves along the x axis2012-11-21
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    Are you doing this in a calculus, or a pre-calculus context? It makes a difference in the interpretation of the problem. If you really mean what you wrote in your last comment, you’re doing it in a pre-calculus context, and the answer that you accepted may not be the one that you want.2012-11-21

2 Answers 2

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In your case you are given $f(0) = 24$. This gives us that $$b a^0 = 24 \implies b = 24$$

You are also given that the growth rate is $0.17$.

Growth rate is typically defined as $$\dfrac1f\dfrac{df}{dx}$$ Since $f(x) = 24 a^x$, we have that $\dfrac{df}{dx} = 24 a^x \log(a)$. Hence, growth rate is $$\dfrac1f\dfrac{df}{dx} = \dfrac{24 a^x \log(a)}{24 a^x} = 0.17$$ This gives us that $a = e^{0.17}$. Hence, $$f(x) = 24e^{0.17x}$$

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    genius, master of math - thanks alot!2012-11-21
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First use the fact that $f(0)=24$: since $f(x)=ba^x$, $24=f(0)=ba^0=b\cdot1=b$, and we now know that $f=24a^x$. Now we use the $17$% growth rate to determine $a$: to get an increase of $17$% with each unit increase in $x$, you need to multiply by $1.17$ ($117$%) every time $x$ increases by $1$, so $a=1.17$, and $f(x)=24(1.17^x)$.