Write an exponential function y = abx for a graph that includes (–4, 72) and (–2, 18).

Step-by-step explanation:

* Lets revise the meaning of exponential function

- The form of the exponential function is ,

  where a ≠ 0, b > 0 ,  b ≠ 1, and x is any real number

- It has a constant base b

- It has a variable exponent x

- To solve an exponential equation, take the log or ln of both sides,  

 and solve for the variable

* Lets solve the problem

∵ y = a(b)^x is an exponential function

∵ Its graph contains the point (-4 , 72) and (-2 , 18)

- Lets substitute x and y by the coordinates of these points

# Point (-4 , 72)

∵

∵ x = -4 and y = 72

∴

- The change any power from -ve to +ve reciprocal the base of

 the power (

∴

- By using cross multiplication

∴ ⇒ (1)

# Point (-2 , 18)

∵ x = -2 and y = 18

∴

∴

- By using cross multiplication

∴ a = 18b² ⇒ (2)

- Equate the two equations (1) and (2)

∴

- Divide both sides by 18b²

∵

∵

∴ 4b² = 1 ⇒ divide both sides by 4

∴ ⇒ take square root for both sides

∴ b = √0.25 = 0.5

- Lets substitute the value ob b in equation (1) or (2) to find a

∵ a = 18b²

∵ b² = 0.25

∴ a = 18(0.25) = 4.5

- Lets substitute the values of a and b in the equation

∴

- We can write it using fraction

∴

ANSWER

EXPLANATION

Let the exponential function be

Since the graph includes (–4, 72), it must satisfy this equation.

Multiply both sides by b⁴ .

This implies that,

The graph also includes (-2,18).

We substitute this point also to get:

Multiply both sides by b²

We equate (1) and (2) to obtain:

Multiply both sides by

Or

Put b=½ into equation (2).

Therefore the equation is

the surface area is 124cm²

step-by-step explanation:

length: 4cm

width: 9cm

height:   2cm