DETERMINE 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.

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.

Decide whether the table represents a linear or exponential function. Then, write the function formula.

Problem 1 :

Solution :

To get values of y, we have to add 3. Then, it is linear function.

Linear function will be in the form of y = mx + b

Slope(m) = 3 and y-intercept (b) = 2

So, the required function is y = 3x + 2

Problem 2 :

Solution :

Since the multiplication factor is the same and it is greater than 1, it is exponential growth function.

Exponential function will be in the form of y = a(b)^x

Initial value (a) = 3

Multiplication factor (b) = 2

So, the required exponential function is y = 3(2)^x

Problem 3 :

Solution :

Observing the values of y,

10/2 ==> 5

5/2 ==> 2.5

Since the multiplication factor is 1/2, that is lesser than 1. It should be exponential decay function

Exponential function will be in the form of y = a(b)^x

Initial value (a) = 10

Multiplication factor (b) = 1/2

So, the required exponential function is y = 10(1/2)^x

Identify the function as linear or exponential and determine the slope or growth factor. Write the rule for each function and sketch a graph labeling the y-intercept.

Problem 4 :

Solution :

Since the multiplication factor is same and it is greater than 2. It is exponential growth function.

Multiplication factor = 2

Exponential function will be in the form of y = a(b)^x

Initial value (a) = 1

So, the required exponential function is y = 1(2)^x

Problem 5 :

Solution :

By observing the values of y, it is added by 2.

-4 + 2 ==> -2

-2 + 2 ==> 0

0 + 2 ==> 2

y = -4x + 2

It is a linear function .

Problem 6 :

Solution :

By observing the values of y, it is multiplied by 2

(1/16) x 2 ==> 1/8

(1/8) x 2 ==> 1/4

(1/4) x 2 ==> 1/2

(1/2) x 2 ==> 1

Multiplication factor = 2

a = 1/4 and b = 2

y = (1/4)(2)^x

It is a exponential function.

Problem 7 :

Solution :

By observing the values of y, it is added by 2

-2 + 2 ==> 0

0 + 2 ==> 2

2 + 2 ==> 4

y-intercept = 2

y = 2 + 2x

It is a linear function.