SOLUTION: write an exponential function through (0, -5) and (-3, -4)

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Question 424108: write an exponential function through (0, -5) and (-3, -4)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The general exponential function has the form:
y+=+a%2Ab%5Ex (with b < 0)
For the specific exponential function whose graph passes through the two given points we have to find the specific values of "a" and "b".

So we have 2 unknowns to find. For this we need two equations. Since the given points are supposed to be on the graph, they must fit the equation. So
-5+=+a%2Ab%5E%280%29%5D%5D%5D%0D%0Aand%0D%0A%7B%7B%7B-4+=+a%2Ab%5E%28-3%29
With these two equations we should be able to find a and b.

In the first equation, since b is not zero, b%5E0+=+1. So it simplifies to:
-5 = a
Already we have found one of the two values we need. To find b we just use the newly found value for a and the second equation:
-4+=+%28-5%29%2Ab%5E%28-3%29
Since it is easier to work with positive exponents we will replace the right side with its positive exponent equivalent:
-4+=+%28-5%29%2A%281%2Fb%5E3%29
which simplifies to:
-4+=+%28-5%29%2Fb%5E3
Next we will eliminate the fraction by multiplying both sides by b%5E3:
-4b%5E3+=+-5
Next we isolate the base and its exponent. Dividing by -4 we get:
b%5E3+=+5%2F4
Now we just find the cube root of each side:
b+=+root%283%2C+4%2F5%29
Last of all we rationalize the denominator:
b+=+root%283%2C+%284%2F5%29%285%5E2%2F5%5E2%29%29
b+=+root%283%2C+%284%2A5%5E2%29%2F5%5E3%29
b+=+root%283%2C+4%2A5%5E2%29%2Froot%283%2C+5%5E3%29
b+=+root%283%2C+100%29%2F5

With the values of a and b that we have found we can finally write the specific exponential equation that passes through the two given points:
y+=+%28-5%29%28root%283%2C+100%29%2F5%29%5Ex