Question

Come up with a new linear function that has a slope that falls in the range −1< m< 0 . Choose two different initial values. For this new linear function, what happens to the function’s values after many iterations? Are the function’s values getting close to a particular number in each case?

Answer 1

5. The slopes approach negative infinity.

#6 was skipped for my teacher.

Question 7 follows a pattern with answers:

3.4

1.2

2.4

3.2

1.6

3.2

1.6 which is the loop

8. Question 8 answers

3.44

1.12

2.24

3.52

0.96

1.92

3.84*

0.32

0.64

1.28

2.56

2.88

2.24

-2.48

3.04

1.92

3.84*

0.32*

Which is the loop.

Problems with answers like this make me wonder if something is wrong with my Algebra 2 teacher, or if he is just mean. One little mistake and you would never find the answer.

9-11 were skipped.

Answer 2

Answer:

1) After many iterations the function will become negative (at some moment)

2) The values are not getting close to a particular number; they will be decreasing indefinetly (toward - ∞)

Justification:

1) You can choose any value of the slope,m, in the interval (-1,0)

2) Use, for example, m = - 1/2

3) Since the slope intercept form of the linear equation is y = mx + b, you have your equation is:

y = (-1/2)x + b

4) That means that for any consecutive itereation the value of the function f, will decrease 1/2.

Suppose that you start with a value y = 1

Then, the next value will be y = 1 - 1/2 = 1/2

The next value will be y = 1/2 - 1/2 = 0

The next value will be y = 0 - 1/2 = -  1/2.

And so on.

The value of y will be decreasing with each iteration. The function f has not limit when x approaches bigger values, it will always become smaller and smalller.

Mathematically, it is said that the limit of function f(x) and x approaches zero is negative infinity.