FROM THE POINTS FIND THE EXPONENTIAL FUNCTION

Find a formula for an exponential function passing through the two points.

Problem 1 :

(0, 6), (3, 750)

Solution:

An exponential function is in the general form

y = a(b)^x

Substitute (0, 6) and (3, 750) for (x, y) in equation

6 = a(b)⁰

6 = a(1)

a = 6

750 = a(b)³

ab³ = 750 ---> (1)

By applying a = 6 in (1)

6b³ = 750

b³ = 125

b = 5

Exponential equation 

y = a(b)^x

y = 6(5)^x

Problem 2 :

(0, 3), (2, 75)

Solution:

An exponential function is in the general form

y = a(b)^x

Substitute (0, 3) and (2, 75) for (x, y) in equation

3 = a(b)⁰

3 = a(1)

a = 3 

75 = a(b)²

ab² = 75 ---> (1)

By applying a = 3 in (1).

3b² = 75

b² = 25

b = 5

Exponential equation 

y = a(b)^x

y = 3(5)^x

Problem 3 :

(0, 2000), (2, 20)

Solution:

An exponential function is in the general form

y = a(b)^x

Substitute (0, 2000) and (2, 20) for (x, y) in equation

2000 = a(b)⁰

2000 = a(1)

a = 2000 

20 = a(b)²

ab² = 20 ---> (1)

By applying a = 2000 in (1).

2000b² = 20

Exponential equation 

y = a(b)^x

Problem 4 :

(0, 9000), (3, 72)

Solution:

An exponential function is in the general form

y = a(b)^x

Substitute (0, 9000) and (3, 72) for (x, y) in equation

9000 = a(b)⁰

9000 = a(1)

a = 9000

72 = a(b)³

ab³ = 72 ---> (1)

By applying a = 9000 in (1).

9000b³ = 72

Exponential equation 

y = a(b)^x

Problem 5 :

Solution:

An exponential function is in the general form

y = a(b)^x

Substitute (-1, 3/2) and (3, 24) for (x, y) in equation

24 = a(b)³

ab³ = 24 ---> (1)

By applying a = 3b/2 in (1).

If b = 2,

Exponential equation 

y = a(b)^x

y = 3(2)^x

Problem 6 :

Solution:

An exponential function is in the general form

y = a(b)^x

Substitute (-1, 2/5) and (1, 10) for (x, y) in equation

10 = a(b)¹

ab = 10 ---> (1)

By applying a = 2b/5 in (1).

If b = 5,

Exponential equation 

y = a(b)^x

y = 2(5)^x

Problem 7 :

(-2, 6), (3, 1)

Solution:

An exponential function is in the general form

y = a(b)^x

Substitute (-2, 6) and (3, 1) for (x, y) in equation

Applying (-2, 6)

6 = a(b)^-2

6 = a/b²

6b² = a

1 = ab³

ab³ = 1 ---> (1)

By applying a = 6b² in (1).

(6b²)b³ = 1

6b⁵ = 1

b⁵ = 1/6

Exponential equation 

y = a(b)^x

Problem 8 :

(-3, 4), (3, 2)

Solution:

An exponential function is in the general form

y = a(b)^x

Substitute (-3, 4) and (3, 2) for (x, y) in equation

4 = a(b)^-3

2 = ab³ 

ab³ = 2 ---> (1)

By applying a = 4b³ in (1).

(4b³)b³ = 2

4b⁶ = 2

Exponential equation 

y = a(b)^x

Problem 9 :

(3, 1), (5, 4)

Solution:

An exponential function is in the general form

y = a(b)^x

Substitute (3, 1) and (5, 4) for (x, y) in equation

1 = a(b)³

ab³ = 1 ---> (1)

4 = a(b)⁵

ab⁵ = 4 ---> (2)

Solve for a and b.

If b = 2,

ab³ = 1

a(2)³ = 1

8a = 1

a = 1/8

Exponential equation 

y = a(b)^x