
Dannielle G.
asked 02/10/21What is an exponential function in the form y=ab^x that passes through the points (1,12) and (1,36)
I really need help ASAP
1 Expert Answer

Logan M. answered 02/10/21
UC Santa Cruz Grad Student in Physics for Math and Physics tutoring
There is no exponential that goes though both (1, 12) and (1, 36). You would need a vertical line, i.e. b = Infinity.
But, in general, if you have an exponential that goes through (x0, y0) and (x1, y1), then:
y0 = f(x0) = a * b^x0
y1 = f(x1) = a * b^x1
then we can find b using:
y0 / y1 = f(x0) / f(x1) = (a * b^x0) / (a * b^x1) = b^x0 / b^x1 = b^(x0 - x1).
That is,
y0 / y1 = b^(x0 - x1)
Taking the log of both sides (say the log base e or the natural log), we find:
ln(y0 / y1) = (x0 - x1) * ln(b)
Solving for ln(b):
ln(b) = ln(y0 / y1) / (x0 - x1)
and exponentiating:
b = e^(ln(y0 / y1) / (x0 - x1))
or
--------------------------------------------
|| b = (y0/y1)^[ 1 / (x0 - x1)] ||
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Then, to find a, use:
y0 = a * b^x0 = a * (y0/y1)^[ x0 / (x0 - x1)]
And thus,
a = y0 / { (y0/y1)^[ x0 / (x0 - x1)] }
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|| a = y0 * (y0/y1)^[ -x0 / (x0 - x1)]. ||
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Mark M.
02/10/21