SOLUTION: write an exponential function whose graph passes through the points (0,4) and (-2,100)

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Question 391614: write an exponential function whose graph passes through the points (0,4) and (-2,100)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The general form for an exponential function is:
f%28x%29+=+a%2Ab%5Ex
where "a" and "b" are non-zero constants. So the task before us is to figure out the specific "a" and "b" that will cause the graph to pass through the two given points.

If the graph passes through (0, 4) then when we use an input of 0 for the function we should get 4 as the output. IOW: f(0) = 4. Substituting 0 and 4 into the general equation we get:
4+=+a%2Ab%5E0
Since b%5E0 equals 1 this becomes:
4 = a*1
or
4 = a
Already we have our "a".

If the graph passes through (-2, 100) then when we use an input of -2 for the function we should get 100 as the output. IOW: f(-2) = 100. Substituting -2 and 100 (and our "a" (4) into the general equation we get:
100+=+%284%29%2Ab%5E%28-2%29
Dividing both sides by 4 we get:
25+=+b%5E%28-2%29
Next we need to find a way to change the exponent on "b" to a 1. The way to do this is to raise both sides of the equation to the reciprocal of -2 power. The reasons we do this are:
  • Raising a power to a power means we will multiply the exponents.
  • When multiplying reciprocals the answer is always a 1!

The reciprocal of -2 is -1/2. So now we have:
%2825%29%5E%28%28-1%2F2%29%29+=+%28b%5E%28-2%29%29%5E%28%28-1%2F2%29%29
On the right side we get b%5E1 or "b" just as we planned. All we have to do is simplify the left side. If you have trouble with negative and/or fraction exponents I find it helpful to factor the exponent in a certain way:
  • If the exponent is negative, factor out a -1.
  • If the exponent is fractional and the numerator is not a 1, factor out the numerator, (For example, factor an exponent like 3%2F4 into 3%2A%281%2F4%29).

Factoring our exponent this way we get:
%2825%29%5E%28%28%28-1%29%2A%281%2F2%29%29%29+=+b
Looking at the factors of the exponent we see a -1 and a 1/2. The -1 (as an exponent) tells us that a reciprocal will be found. And the 1/2 (as an exponent) means a square root will be found. And since multiplication is Commutative, we can do these operations in any order we choose! Since finding a square root of 25 seems easier than the reciprocal, I choose to start with that. Then we will finish with a reciprocal:
%2825%29%5E%28%28%281%2F2%29%2A%28-1%29%29%29+=+b
%2825%5E%28%281%2F2%29%29%29%5E%28-1%29+=+b
%285%29%5E%28-1%29+=+b
1%2F5+=+b

Now that we have our "a" and "b" we can write the desired function:
f%28x%29+=+4%2A%281%2F5%29%5Ex
or (if you prefer decimals):
f%28x%29+=+4%2A%280.2%29%5Ex