# What are $a$ and $b$?

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?

-
What happens when you substitute $x=0$ into the given expression for $f(x)$? – Brad Nov 21 '12 at 18:52
Do you know some calculus? – André Nicolas Nov 21 '12 at 18:52
As it says, f(0) = 24 - all that i know is above – Frederik Witte Nov 21 '12 at 18:55
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'? – Amzoti Nov 21 '12 at 18:55
17% is the growth of the function, it's expotential, it's raising with 17% for each time it moves along the x axis – Frederik Witte Nov 21 '12 at 18:56

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, $$f(x) = 24e^{0.17x}$$
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 $f(x)=24(1.17^x)$.