Identify the initial amount and the growth factor b in the exponential function. y= 5(0.5)^x

Question:

Identify the initial amount and the growth factor b in the exponential function.

{eq}y= 5(0.5)^x {/eq}

Growth/Decay:

An exponential function is of the form {eq}r= ab^x {/eq}.

Here {eq}a {/eq} is the initial value of the function.

(i) If {eq}b>1 {/eq} then the function represents the growth and the growth rate is given by {eq}r {/eq} which is obtained by solving {eq}1+r=b {/eq}.

(ii) If {eq}b<1 {/eq} then the function represents the decay and the decay rate is given by {eq}r {/eq} which is obtained by solving {eq}1-r=b {/eq}.

Answer and Explanation:

The given exponential equation is,

$$y= 5(0.5)^x $$

Comparing this with {eq}y=ab^x {/eq}, we get

$$\begin{align} \text{The initial amount, }a &=5 \\ b&=0.5 \end{align} $$

Here, {eq}b=0.5 <1 {/eq}.

So the given equation represents the exponential decay and the decay factor is obtained by solving

$$1-r=b \\ 1-r =0.5 \\ \text{Subtracting 1 from both sides}, \\ -r=-0.5 \\ \text{Multiply both sides by -1},\\ r= 0.5 $$

Therefore, the initial amount is, {eq}a= \boxed{\mathbf{5}} {/eq}

and the decay factor is, {eq}r= \boxed{\mathbf{0.5}} {/eq}.


Learn more about this topic:

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Exponential Growth vs. Decay

from Math 101: College Algebra

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