Answer:
[tex]y=4.5(0.5)^{x}[/tex]
Step-by-step explanation:
* Lets revise the meaning of exponential function
- The form of the exponential function is [tex]y=ab^{x}[/tex],
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)
∵ [tex]y=ab^{x}[/tex]
∵ x = -4 and y = 72
∴ [tex]72=ab^{-4}[/tex]
- The change any power from -ve to +ve reciprocal the base of
the power ([tex]p^{-n}=\frac{1}{p^{n}}[/tex]
∴ [tex]72=\frac{a}{b^{4}}[/tex]
- By using cross multiplication
∴ [tex]a=72b^{4}[/tex] ⇒ (1)
# Point (-2 , 18)
∵ x = -2 and y = 18
∴ [tex]18=ab^{-2}[/tex]
∴ [tex]18=\frac{a}{b^{2}}[/tex]
- By using cross multiplication
∴ a = 18b² ⇒ (2)
- Equate the two equations (1) and (2)
∴ [tex]72b^{4}=18b^{2}[/tex]
- Divide both sides by 18b²
∵ [tex]\frac{72b^{4}}{18b^{2}}=4b^{4-2}=4b^{2}[/tex]
∵ [tex]\frac{18b^{2}}{18b^{2}}=(1)b^{2-2}=(1)b^{0}=(1)(1)=1[/tex]
∴ 4b² = 1 ⇒ divide both sides by 4
∴ [tex]b^{2}=\frac{1}{4}=0.25[/tex] ⇒ 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 [tex]y=ab^{x}[/tex]
∴ [tex]y=4.5(0.5)^{x}[/tex]
- We can write it using fraction
∴ [tex]y=\frac{9}{2}(\frac{1}{2})^{x}[/tex]
ANSWER
[tex]y = \frac{9}{2} ( { \frac{1}{2} })^{x}[/tex]
EXPLANATION
Let the exponential function be
[tex]y = a {b}^{x} [/tex]
Since the graph includes (–4, 72), it must satisfy this equation.
[tex]72= a { b}^{ - 4}[/tex]
Multiply both sides by b⁴ .
This implies that,
[tex]a = 72 {b}^{4} ...1[/tex]
The graph also includes (-2,18).
We substitute this point also to get:
[tex]18=a {b}^{ - 2} [/tex]
Multiply both sides by b²
[tex]a = 18 {b}^{2} ...(2)[/tex]
We equate (1) and (2) to obtain:
[tex]72 {b}^{4} = 18 {b}^{2} [/tex]
Multiply both sides by
[tex] \frac{72 {b}^{4} }{ {18b}^{4} } = \frac{18 {b}^{2} }{18 {b}^{4} } [/tex]
[tex]4 = \frac{1}{ {b}^{2} } [/tex]
Or
[tex]{2}^{ 2} = ( \frac{1}{b} )^{2} [/tex]
[tex] \frac{1}{b} = 2[/tex]
[tex]b = \frac{1}{2} [/tex]
Put b=½ into equation (2).
[tex]a = 18 {( \frac{1}{2} })^{2} [/tex]
[tex]a = \frac{18}{4} [/tex]
[tex]a = \frac{9}{2} [/tex]
Therefore the equation is
[tex]y = \frac{9}{2} ( { \frac{1}{2} })^{x} [/tex]
