Question

Can you tell me if this is right. I am confused, please see attachment.

Answer 1

All the function as exponential and they can be written in the form:

$y=A\cdot b^t$

When written in thif form, we can check if the function describe an exponential growth or decay by checking "b".

$\begin{gathered} if\, 01\to exponential\, growth \end{gathered}$

Let's start with the two that are already in this form:

$y=0.1(1.25)^t$

Here, b = 1.25. Since b > 1, this describe an exponential growth.

$y=426(0.98)^t$

b = 0.98. Since 0 < b < 1, this describe an exponential decay.

Now, this one is almost in this form:

$y=2050(\frac{1}{2})^t$

We just need to turn b = 1/2 to decimal to be sure. 1/2 is 0.5, so b = 0.5. Since 0 < b < 1, this describe an exponential decay.

This one, we need to evaluate the expression inside parenthesis:

$y=100(1-\frac{1}{2})^t$

Here, b is:

$b=1-\frac{1}{2}=1-0.5=0.5$

Since 0 < b < 1, this describe an exponential decay.

Lastly,, we have the following:

$y=((1-0.03)^{\frac{1}{2}})^{2t}$

First, le'ts make the substraction:

$y=((0.97)^{\frac{1}{2}})^{2t}$

Now, notice that when we have a multiplication in an exponent, we can do the following:

$a^{c\cdot d}=(a^c)^d$

Thus, we can do the contrary. So,

$y=((0.97)^{\frac{1}{2}})^{2t}=(0.97)^{\frac{1}{2}\cdot2t}=(0.97)^t$

Notice that we don't have anything in the place of "A", but this means that A = 1. This doesn't change b, that is, in this case equal to 0.97. Since 0 < b < 1, this describe an exponential decay.

So, the function that describe exponential growth is:

$0.1(1.25)^t$

And the functions that describe exponential decay are:

$\begin{gathered} y=((1-0.03)^{\frac{1}{2}})^{2t} \\ y=2050(\frac{1}{2})^t \\ y=426(0.98)^t \\ y=100(1-\frac{1}{2})^t \end{gathered}$