Final answer:
The exponential function that passes through the given points is found by setting up equations based on the general exponential form and solving for the base and the coefficient. The resulting function is y = 2(4)^x.
Step-by-step explanation:
To find the exponential function that passes through the points (-3, 1/32) and (3, 128), we can use the general form y = abx. For simplicity, let's call the y-values from the points y1 and y2, and the x-values x1 and x2 respectively.
Firstly, we have:
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- y1 = abx1 => 1/32 = ab-3
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- y2 = abx2 => 128 = ab3
By dividing the second equation by the first, we get:
(128) / (1/32) = (ab3) / (ab-3)
This simplifies to:
4096 = b3 / b-3 = b6
So, b = 4 since 46 = 4096.
Substituting b = 4 into one of the original equations to get a:
1/32 = a(4)-3 => a = 1/32 × 43 => a = 1/32 × 64 => a = 2
The exponential function is therefore:
y = 2(4)x
This answers the student's question with the exponential equation matching the given points.