Question

The first four terms of a sequence are shown below.

8, 5, 2, -1

Which of the following functions best defines this sequence?

f(1) = 8, f(n + 1) = f(n) + 3; for n _ 1

f(1) = 8, f(n + 1) = f(n) - 5; for n _ 1

f(1) = 8, f(n + 1) = f(n) + 5; for n _ 1

f(1) = 8, f(n + 1) = f(n) - 3; for n _ 1

Which of the following statements best describes the effect of replacing the graph of y = f(x) with the graph of y = f(x) + 20?

The graph of y = f(x) will shift left 20 units.

The graph of y = f(x) will shift down 20 units.

The graph of y = f(x) will shift up 20 units.

The graph of y = f(x) will shift right 20 units.


Every week Annie plants 2 saplings in her garden. The function below shows the total number of saplings f(w) in her garden after w weeks.

f(w) = 2w + 3

Which statement best describes the y-intercept of the graph of the function?

It is equal to 3 and represents the original number of saplings in Annie's garden.

It is equal to 2 and represents the original number of saplings in Annie's garden.

It is equal to 3 and represents the number of saplings in Annie's garden at the end of the first week.

It is equal to 5 and represents the number of saplings in Annie's garden at the end of the first week.

Answer 1

1.) f(1) = 8, f(n + 1) = f(n) - 3; for n ≥ 1

2.) The graph of y = f(x) will shift up 20 units.

3.) It is equal to 3 and represents the original number of saplings in Annie's garden.

Answer 2

Part 1) The first four terms of a sequence are shown below.

Let

`f(1)=8\\ f(2)=5\\ f(3)=2\\ f(4)=-1`

we know that

`f(2)-f(1)=5-8\\ f(2)-f(1)=-3\\ f(2)=f(1)-3`

`f(3)-f(2)=2-5\\ f(3)-f(2)=-3\\ f(3)=f(2)-3`

`f(4)-f(3)=-1-2\\ f(4)-f(3)=-3\\ f(4)=f(3)-3`

So

`f(n+1)=f(n)-3`

therefore

the answer part 1) is the option

`f(1) = 8, f(n + 1) = f(n) - 3; for. n\geq 1`

Part 2) Which of the following statements best describes the effect of replacing the graph of y = f(x) with the graph of y = f(x) + 20?

Let

(a,b)--------->represent a point in the graph of f(x)

we know that

In the new function f(x)+d

(a,b)-----------> become-------> (a,b+d)

So

The new function f(x)+d shift up by d

therefore

in the new function
`f(x)+20`-------> the graph of f(x) shift up by
`20`

the answer part 2) is the option

The graph of y = f(x) will shift up
`20` units.

Part 3) Every week Annie plants 2 saplings in her garden. The function below shows the total number of saplings f(w) in her garden after w weeks.

`f(w) = 2w + 3`

we know that

The y-intercept is when the value of w is equal to zero

the value of f(w) for

`w=0`

represent the original number of saplings in Annie's garden

so

For

`w=0\\ f(0)=2*0+3\\ f(0)=3`

therefore

the answer is the option

It is equal to
`3` and represents the original number of saplings in Annie's garden.