Answer:
- f(x) = 2·5^x . . . b = 5; a = 2
- f(x) = (9/2)(2/3)^x . . . b = 2/3; a = 9/2
- f(x) = (1/2)4^x . . . b = 4; a = 1/2
Step-by-step explanation:
Actually, the lesson is giving you wrong information. The exponential function written in the example does not work for the point (4, 24).
Here's another way to look at it.
Given
two points: (x1, y1) and (x2, y2)
Solution
The exponential function can be written as ...
y = (y1)(y2/y1)^((x -x1)/(x2 -x1))
Example
For the example points, the function would be ...
y = (6)(24/6)^((x-2)/(4-2)) = 6·4^(x/2 -1) = (6/4)(4^(1/2))^x = (3/2)2^x
Check
For x=2, y=(3/2)2^2 = 6. For x=4, y = (3/2)2^4 = 24.
You will notice that we figure out "b" and "a" as a part of the process of simplifying the equation. The equation would work perfectly well without simplification.
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1. Given points (2, 50) and (3, 250), the equation is ...
f(x) = 50(250/50)^((x -2)/(3 -2)) = 50·5^(x -2) = (50/25)5^x
f(x) = 2·5^x . . . b = 5; a = 2
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2. Given points (1, 3) and (2, 9/2), the equation is ...
f(x) = (3)(3/(9/2))^((x -1)/(2 -1)) = (3)(2/3)^(x -1) = (3/(2/3))(2/3)^x
f(x) = (9/2)(2/3)^x . . . b = 2/3; a = 9/2
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3. Given points (3, 32) and (4, 128), the equation is ...
f(x) = (32)(128/32)^((x -3)/(4 -3)) = (32)4^(x -3) = (32/64)4^x
f(x) = (1/2)4^x . . . b = 4; a = 1/2
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Additional comments
The usual rules of exponents are useful in this:
a^(b+c) = (a^b)(a^c)
a^(bc) = (a^b)^c
a^-b = 1/a^b
