Find the exponential model of best fit for the points (−3, 5),(1, 12),(5, 72),(7, 137)
To find the exponential model of best fit for the given points, we need to assume that the relationship between the x-values and the y-values follows the exponential form of y = a * b^x.
Let's substitute the given points into this equation to create a system of equations we can solve for a and b:
1) (-3, 5):
5 = a * b^-3
2) (1, 12):
12 = a * b^1
3) (5, 72):
72 = a * b^5
4) (7, 137):
137 = a * b^7
We can rewrite these equations as:
b^-3 = 5/a ---- [Equation 1]
b = 12/a ---- [Equation 2]
b^5 = 72/a ---- [Equation 3]
b^7 = 137/a ---- [Equation 4]
Now, let's divide Equation 3 by Equation 1:
(b^5)/(b^-3) = (72/a)/(5/a)
b^8 = (72/a)*(5/a)
b^8 = 360/a^2
And let's divide Equation 4 by Equation 2:
(b^7)/b = (137/a)/(12/a)
b^6 = (137/12)
b^6 = 137/12
Now, let's substitute b^6 = 137/12 into the equation b^8 = 360/a^2:
(137/12)^4 = 360/a^2
(137/12)^4 * a^2 = 360
a^2 = 360 / (137/12)^4
a^2 ≈ 0.473
Taking the square root of both sides, we find:
a ≈ √0.473
a ≈ 0.687
Now, let's substitute this value of a into Equation 2:
b = 12/a
b = 12/0.687
b ≈ 17.46
Therefore, the exponential model of best fit for the given points is y ≈ 0.687 * 17.46^x.