# DETERMINING IF THE TABLE REPRESENTS LINEAR OR EXPONENTIAL FUNCTION

Linear function :

If the growth or decay involves increasing or decreasing by a fixed number(constant difference), then it should be a linear function.

y = mx + b

b is y-intercept and m is slope.

Arithmetic sequence must be a linear function.

Exponential function :

If the growth or decay involves using multiplication, then it should be a exponential function.

y = a(b)x

a is starting value

b is multiplication factor, either growth or decay.

If b > 0 for growth and 0 < b < 1 when it is decay.

Geometric sequence must be a exponential function.

The following functions are either linear or exponential. Which is it? Justify your answer.

Problem 1 :

Solution :

Changes in input :

2 - (-3) ==> 5, 7 - 2 ==> 5

So, the change inputs is same.

Finding the common difference :

3 -1 ==> 2

9 - 3 ==> 6

Since the difference is not same, it is not arithmetic and not linear.

Finding the common ratio :

1 x 3 ==> 3

3 x 3 ==> 9

Since the common ratio is the same, it must be a exponential function. f changes proportionality by a ratio of 3.

Problem 2 :

Solution :

Changes in input :

6 - 2 ==> 4, 10 - 6 ==> 4

So, the change inputs is same.

Finding the common difference :

9 -12 ==> -3

6 - 9 ==> -3

Since the difference is same, it is a arithmetic and linear.

f changes at the constant rate of -3.

Problem 3 :

Solution :

Changes in input :

5 - 1 ==> 4, 9 - 5 ==> 4

So, the change inputs is same.

Finding the common difference :

6 - 2 ==> 4

18 - 6 ==> 12

Since the difference is not same, it is not arithmetic and not linear.

Finding the common ratio :

2 x 3 ==> 6

6 x 3 ==> 18

Since the common ratio is the same, it must be a exponential function. f changes proportionality by a ratio of 3.

Problem 4 :

Solution :

Changes in input :

-2 - (-4) ==> -2 + 4 ==> 2

0 - (-2) ==> 0 + 2 ==> 2

So, the change inputs is same.

Finding the common difference :

1 - 5 ==> -4

-3 - 1 ==> -4

Since the difference is same, it is arithmetic and linear.

Problem 5 :

Solution :

Changes in input :

10 - 7 ==> 3, 13 - 10 ==> 3

So, the change inputs is same.

Finding the common difference :

105 - 5 ==> 100

205 - 105 ==> 100

Since the difference is same, it is arithmetic and linear.

Problem 6 :

Solution :

Changes in input :

20 - 11 ==> 9, 29 - 20 ==> 9

So, the change inputs is same.

Finding the common difference :

4 - 8 ==> -4

2 - 4 ==> -2

Since the difference is not same, it is not arithmetic and not linear.

Finding the common ratio :

8 x (1/2) ==> 2

4 x (1/2) ==> 2

Since the common ratio is the same, it must be a exponential function. f changes proportionality by a ratio of 2.

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