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.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM