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?
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?
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 $f(x)=24(1.17^x)$.