To find the exponential model of best fit, we need to express the relationship between the x-values and y-values in the form of an exponential equation.
The general form of an exponential equation is y = a * b^x, where a and b are constants to be determined.
We can start by writing out the equation using the given points:
For the point (-3, 5): 5 = a * b^(-3)
For the point (1, 12): 12 = a * b^(1)
For the point (5, 72): 72 = a * b^(5)
For the point (7, 137): 137 = a * b^(7)
We now have a system of four equations with two unknowns (a and b). We can solve this system to find the values of a and b.
One way to solve this system is by using a method called substitution. Begin by isolating a in terms of b from the first equation:
5 = a * b^(-3)
a = 5 / b^(-3)
a = 5 * b^3
Now substitute this expression for a in the other three equations:
12 = (5 * b^3) * b^(1)
72 = (5 * b^3) * b^(5)
137 = (5 * b^3) * b^(7)
Rearranging the last two equations, we get:
12 = 5b^4
72 = 5b^8
Now we have two equations with the same base raised to different powers. We can equate the two expressions:
5b^4 = 5b^8
Divide both sides by 5:
b^4 = b^8
Now we can take the fourth root of both sides:
(b^4)^(1/4) = (b^8)^(1/4)
b = b^2
We have found that b equals 1 or -1.
If b = 1, then a = 5 * 1^3 = 5.
If b = -1, then a = 5 * (-1)^3 = -5.
Therefore, we have two possible exponential models of best fit:
1. y = 5 * 1^x, which simplifies to y = 5.
2. y = -5 * (-1)^x, which simplifies to y = -5 when x is even, and y = 5 when x is odd.
Both models fit the given points: (-3, 5), (1, 12), (5, 72), and (7, 137). However, note that using the second model may not be mathematically preferable since it alternates between two values based on the parity of x.
Rounding the values to the nearest hundredth, we have:
1. y = 5
2. y = 5 when x is odd, and y = -5 when x is even.
Find the exponential model of best fit for the points (- 3, 5), (1, 12), (5, 72), (7, 137) Explain how you got your answer Round values to the nearest hundredth
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