Question

consider the following graph on an exponential function modeling the geometric sequence 1, 3, 9, 27, ...which of the following statements are valid based on the graph select all correct answers in choices ?

Answer 1

Consider the exponential growth function

`y=a(b)^x`

where b is the growth factor.

Consider the points (0,1) and (-1,1/3).

Substitute x=0 and y=1, we get

`1=a(b)^0`

`1=a`

Substitute x=-1, y=1/3, and a=1 in the equation, we get

`(1)/(3)=1(b)^(-1)`

`(1)/(3)=(1)/(b)`
`b=(1)/((1)/(3))`

`b=3`

When the coordinates (0,1) and (-1,1/3) the growth factor is r=3.

The first option is correct.

Consider the points (1,3) and (2,9).

Substitute x=1 and y=3, we get

`3=a(b)^1`

`a=(3)/(b)`

Substitute x=2, y=9 and a=3/(1+r), we get

`9=(3)/(b)(b)^2`

`(9)/(3)=(b)`

`b=3`

when the coordinated (1,3) and (2,9) are considered the growth factor is 3.

The second option is incorrect.

The growth factor does not change if we choose different points.

So the correct options are

1. When the coordinates (0,1) and (-1,1/3) are considered r=1/(1/3) which simplifies to 3.

3. when the coordinated (3.27) and (2,9) are considered r=27/9 which simplifies to 3.

6. when the coordinated (1,3) and (2,9) are considered r=9/3 which simplifies to 3.