Question

Algebra 2 Help please!! I will give a medal. IN WHAT DIFFERENT WAYS CAN EXPONENTIAL FUNCTIONS BE IDENTIFIED?

Answer 1

Do you have a multiply choice to the question, is this a question that you are supposed to address in your own words, or are you just trying to learn about exponential functions on your own?

Answer 2

@SolomonZelman It is just a question that I have to answer. Can you please help me?

Answer 3

I will assume you know what exponent is.

Answer 4

Yes, I do @SolomonZelman

Answer 5

So, for example:
\(\color{#000000 }{ \displaystyle 2^1=2 }\)
\(\color{#000000 }{ \displaystyle 2^2=2\cdot 2 }\) (2 times)
\(\color{#000000 }{ \displaystyle 2^3=2\cdot 2\cdot 2 }\) (3 times)
\(\color{#000000 }{ \displaystyle 2^4=2\cdot 2\cdot 2 \cdot 2}\) (4 times)
\(\color{#000000 }{ \displaystyle 2^x=2~~~{....}~~~ 2 \cdot 2}\) (x times)

So the more times you multiply times 2 our output will increase.
And this is not just when you multiply times 2.
Every time you multiply times a number that is
greater than 1 (let's call it b), you get a greater result.

correct?

Answer 6

So, if you make a function,
\(\color{#000000 }{ \displaystyle y=b^x}\)

where b is a number greater than 1, than as x increases, the output (y) would also increase.

Answer 7

what does the output mean ? @SolomonZelman

Answer 8

output is the result you get for y

Answer 9

input is the x value that you plug into the function

Answer 10

And any function in a form of
\(\color{#000000 }{ \displaystyle y=b^x}\)
(where b is some number greater than 1)

(or in a form of \(\color{#000000 }{ \displaystyle y=a(b)^x}\), when
the function is multiplied times a
scale factor "a")

THEN, your output will increase when input increases.

Answer 11

So functions like this
\(\color{#000000 }{ \displaystyle y=b^x}\)
\(\color{#000000 }{ \displaystyle y=a(b)^x}\)

(where b is greater than 1)

is called "exponential growth"

Answer 12

When x is bigger, you multiply times b more times.
And since be is greater than 1, multiplying times 1
will yield larger and larger results.

This is why such functions are called exponential growth

Answer 13

are you getting what I am saying? any confusions so far?

Answer 14

I am getting it so far but I am re-reading it all. @SolomonZelman

Answer 15

Ok, sure... tell me when you are done, and if you have any questions ask, if not I'll proceed

Answer 16

So, exponential growth is one of the ways to identified exponential functions, right? Are there more ways?

Answer 17

And If I graph exponential growth that will be a not straight line, right? @SolomonZelman

Answer 18

No it won't although it will tend to be a vertical line as x grows, because its rate of change is increasing.

Answer 19

it's slow will grow as large as you wish, and will "approach infinity"

Answer 20

Which one is smaller
\(\color{#000000 }{ \displaystyle 0.5^1 }\) or \(\color{#000000 }{ \displaystyle 0.5^4 }\)

and tell me why?

Answer 21

the one with the exponent of 4 is the bigger because 0.5 will get larger as I times it 4 times so 0.5^1 will be the smaller. Am I right? That's what my understanding from what you told me @SolomonZelman

Answer 22

0.5 is same as 1/2

Answer 23

when you multiply 1 times 1/2 you will get a result that is smaller, wouldn't you?

Answer 24

What is smaller,
\(\color{#000000 }{ \displaystyle 1\times (1/2) }\) or 1 ?

Answer 25

1x(1/2) is smaller @SolomonZelman

Answer 26

yes

Answer 27

So I will list the products in order, and you will tell me what happens to them.

Answer 28

\(\color{#000000 }{ \displaystyle 1\times (1/2) }\)
\(\color{#000000 }{ \displaystyle 1\times (1/2)\times (1/2) }\)
\(\color{#000000 }{ \displaystyle 1\times (1/2)\times (1/2) \times (1/2) }\)
\(\color{#000000 }{ \displaystyle 1\times (1/2)\times (1/2) \times (1/2)\times (1/2) }\)
\(\color{#000000 }{ \displaystyle 1\times (1/2)\times (1/2) \times (1/2) \times (1/2)\times (1/2) }\)

Answer 29

the product is increasing?

Answer 30

or is it decreasing?

Answer 31

it is increasing @SolomonZelman

Answer 32

no

Answer 33

it is decreasing.
tell me why is it decreasing

Answer 34

oh its because of the one half

Answer 35

@SolomonZelman

Answer 36

Explain

Answer 37

because it you times 0.5 to 0.5 the results that you will get will be half of the number and it will be the same for the rest @SolomonZelman

Answer 38

Yes

Answer 39

So, if you set,

\(\color{#000000 }{ \displaystyle y=(0.5)^x}\)

then as x grows, what happens to y?
(explain)

Answer 40

It decreases because you are multiplying the x in a half @SolomonZelman

Answer 41

The y decreases, because the bigger x is, the more times you multiply times (1/2), and therefore the more the result becomes smaller.

Answer 42

And in fact, (should be understandable), that

\(\color{#000000 }{ \displaystyle y=b^x}\)

(if b is positive but less than 1)

then, when you multiply times this "b" the more you minimize your result. That means that the bigger the x, the more you multiply times b, and therefore the smaller tha y you have.

Answer 43

So a function is a form
\(\color{#000000 }{ \displaystyle y=b^x}\)
(if: 0>b>1 -- positive b, but smaller than 1)

then as x tends to larger values, the y tends to 0.

And therefore this function is called "exponential decay"

Answer 44

is the exponential decay also part of how to identify an exponential functions? @SolomonZelman

Answer 45

when I wrote 0>b>1, I mean to write 0<b<1

Answer 46

well, yes.

Answer 47

There are two basically

Answer 48

\(\color{#000000 }{ \displaystyle y=b^x}\) or \(\color{#000000 }{ \displaystyle y=a(b)^x}\) when b is greater than 1.
(each time you multiply times b, the result increases, so these two functions are called "exponential growth")

\(\color{#000000 }{ \displaystyle y=b^x}\) or \(\color{#000000 }{ \displaystyle y=a(b)^x}\) when b ispositive, but smaller than 1.
(each time you multiply times b, the result decreases, so these two functions are called "exponential decay"")

Answer 49

\(\color{#000000 }{ \displaystyle y=b^x}\) is same as \(\color{#000000 }{ \displaystyle y=1\cdot (b)^x}\)

so you can write just the
\(\color{#000000 }{ \displaystyle y=a(b)^x}\)
for eac one. you don't need to list both

Answer 50

@SolomonZelman Thank you so much!! It makes it easier for me to understand.