Write an equation of an exponential function that passes through the points (2, 200) and (-1, 102.4)

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Given Points: We are given two points (2, 200) and (-1, 102.4) through which an exponential function passes. The general form of an exponential function is y = ab^x, where a is the initial value and b is the base of the exponential function. We need to find the values of a and b.Create Equation 1: Let's use the first point (2, 200) to create an equation. Substituting x = 2 and y = 200 into the exponential function y = ab^x, we get: 200 = ab^2Create Equation 2: Now, let's use the second point (-1, 102.4) to create another equation. Substituting x = -1 and y = 102.4 into the exponential function y = ab^x, we get: 102.4 = ab^(-1)Solve System of Equations: We now have a system of two equations with two unknowns: 1) 200 = ab^2 2) 102.4 = ab^(-1) We can solve this system by dividing the second equation by the first equation to eliminate the variable a.Divide Equations: Dividing the second equation by the first equation gives us: (102.4 / 200) = (ab^(-1)) / (ab^2) 0.512 = 1/b^3 Now we need to solve for b.Solve for b: To solve for b, we take the cube root of both sides of the equation: b = (0.512)^(1/3) Calculating this gives us: b ≈ 0.8Substitute for b: Now that we have the value of b, we can substitute it back into either of the original equations to solve for a. Let's use the first equation: 200 = ab^2 200 = a(0.8)^2 200 = a(0.64)Solve for a: To solve for a, we divide both sides by 0.64: a = 200 / 0.64 a ≈ 312.5Final Exponential Function: We have found the values of a and b for the exponential function. The equation of the exponential function that passes through the points (2, 200) and (-1, 102.4) is: y = 312.5 * 0.8^x

Write an equation of an exponential function that passes through the points $(2, 200)$ and $(-1, 102.4)$

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Write an equation of an exponential function that passes through the points $(2, 200)$ and $(-1, 102.4)$