Question

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

Answer 1

`y = (9)/(2) ( { (1)/(2) })^(x)`

EXPLANATION

Let the exponential function be

`y = a {b}^(x)`

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

`72= a { b}^( - 4)`

Multiply both sides by b⁴ .

This implies that,

`a = 72 {b}^(4) ...1`

The graph also includes (-2,18).

We substitute this point also to get:

`18=a {b}^( - 2)`

Multiply both sides by b²

`a = 18 {b}^(2) ...(2)`

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

`72 {b}^(4) = 18 {b}^(2)`

Multiply both sides by

`\frac{72 {b}^(4) }{ {18b}^(4) } = \frac{18 {b}^(2) }{18 {b}^(4) }`

`4 = \frac{1}{ {b}^(2) }`

Or

`{2}^( 2) = ( (1)/(b) )^(2)`

`(1)/(b) = 2`

`b = (1)/(2)`

Put b=½ into equation (2).

`a = 18 {( (1)/(2) })^(2)`

`a = (18)/(4)`

`a = (9)/(2)`

Therefore the equation is

`y = (9)/(2) ( { (1)/(2) })^(x)`

Answer 2

`y=4.5(0.5)^(x)`

Explanation:

* Lets revise the meaning of exponential function

- The form of the exponential function is
`y=ab^(x)`,

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)

∵
`y=ab^(x)`

∵ x = -4 and y = 72

∴
`72=ab^(-4)`

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

the power (
`p^(-n)=(1)/(p^(n))`

∴
`72=(a)/(b^(4))`

- By using cross multiplication

∴
`a=72b^(4)` ⇒ (1)

# Point (-2 , 18)

∵ x = -2 and y = 18

∴
`18=ab^(-2)`

∴
`18=(a)/(b^(2))`

- By using cross multiplication

∴ a = 18b² ⇒ (2)

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

∴
`72b^(4)=18b^(2)`

- Divide both sides by 18b²

∵
`(72b^(4))/(18b^(2))=4b^(4-2)=4b^(2)`

∵
`(18b^(2))/(18b^(2))=(1)b^(2-2)=(1)b^(0)=(1)(1)=1`

∴ 4b² = 1 ⇒ divide both sides by 4

∴
`b^(2)=(1)/(4)=0.25` ⇒ 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
`y=ab^(x)`

∴
`y=4.5(0.5)^(x)`

- We can write it using fraction

∴
`y=(9)/(2)((1)/(2))^(x)`