Question

Does each function describe exponential growth or decay? Drag and drop the equations into the boxes to correctly complete the table. Growth Decay y=100(1−12)^t y=0.1(1.25)^t y=((1−0.03)12)^2t y=426(0.98)^t

asked Oct 2, 2018 116k views

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Rakshith G B · Answer 1 · 2018-10-03T10:49:41+0000

Answer:

1. isn't an exponential function

2. growths

3. growths

4. decay

5. growths

Explanation:

The exponential functions has the form: y = k1*k2^(k3*t), where k1, k2 and k3 are a constants, and k2>0. Graph of exponential functions always decrease or always increase. To know if a function growths or decay just evaluate the function in 2 points, for example t = 0 and t = 1, and compare their results.

1.

y = 100*(1−12)^x = 100*(−11)^x

Since (-11) is negative, then y is not an exponential function

2.

y(0) = 0.1*(1.25)^(0) = 0.1

y(1) = 0.1*(1.25)^(1) = 0.125

y(1) > y(0) -> growths

3.

y(0) =((1−0.03)12)^2(0) = 1

y(0) =((1−0.03)12)^2(1) = 135.4896

y(1) > y(0) -> growths

4.

y(0) = 426(0.98)^(0) = 426

y(1) = 426(0.98)^(1) = 417.48

y(1) < y(0) -> decay

5.

y(0) = 2050(12)^(0) = 2050

y(1) = 2050(12)^(1) = 24600

y(1) > y(0) -> growths

Thatismatt · Answer 2 · 2018-10-09T03:00:54+0000

A function
from a set
to a set
is a relation that assigns to each element
in the set
exactly one element
in the set
. The set
is the domain (also called the set of inputs) of the function and the set
contains the range (also called the set of outputs).

$We \ denote \ the \ \mathbf{exponential \ function} \ f \ with \ base \ a \ as: \\ \\ f(x)=a^x \\ \\ where \ a>0, \ a\\eq 1, \ and \ x \ is \ any \ real \ number$ .

1. It isn't an exponential function.

We have the following equation:

y=100(1-12)^t

That can be written as:

y=100(-11)^t

Recall that the definition of exponential functions establishes that:

$We \ denote \ the \ \mathbf{exponential \ function} \ f \ with \ base \ a \ as: \\ \\ f(x)=a^x \\ \\ where \ a>0, \ a\\eq 1, \ and \ x \ is \ any \ real \ number$ .

That is:

$a \ \mathbf{must} \ be \ greater \ than \ 1$

In this problem,
a=-11 , therefore this is not an exponential function.

2. Growth.

The function:

y=0.1(1.25)^t

is an exponential function because is a function of the form
$f(t)=ka^t \\ \\ where \ a>0 \ and \ k \ constant$

So
$k=0.1 \ and \ a=1.25$ . Since
$a \ is \ greater \ than \ 1$ and being raised to the power of
, the function increases. This means that
increases as
increases as illustrated in Figure 1. This represents a growth.

3. Growth.

The function:

y=((1-0.03)12)^(2t) can be written as:

y=11.64^(2t)

and is an exponential function because is a function of the form
$f(t)=a^(bt) \\ \\ where \ a>0 \ and \ b \ constant$

So
$a=11.64 \ and \ b=2$ . Since
$a \ is \ greater \ than \ 1$ and being raised to the power of
, the function increases. As in the previous exercise, this means that
increases as
increases as illustrated in Figure 2. This represents a growth.

4. Decay.

The function:

y=426(0.98)^t

is an exponential function because is a function of the form
$f(t)=ka^t \\ \\ where \ a>0 \ and \ k \ constant$

So
$k=426 \ and \ a=0.98$ . Since
$a \ is \ less \ than \ 1$ and being raised to the power of
, the function decreases. Here this means that
decreases as
increases as illustrated in Figure 3. This represents a decay.

5. Growth.

The function:

y=2050(12)^t

is an exponential function because is a function of the form
$f(t)=ka^t \\ \\ where \ a>0 \ and \ k \ constant$

So
$k=2050 \ and \ a=12$ . Since
$a \ is \ greater \ than \ 1$ and being raised to the power of
, the function increases. So in this function
also increases as
increases as illustrated in Figure 4. This represents a growth.