Find the formula for an exponential function that passes through the two points given. (x, y) = (0, 2100) and (x, y) = (2, 21)
The exponential function `f` has a form
To find `a` and `b,` substitute given x'es.
The first point, x=0 and f(x)=2100:
`2100 = f(0) = a*b^0 = a.` So `a = 2100.`
The second point, x=2 and f(x)=21:
`21 = f(2) = a*b^2 = 2100*b^2.`
Therefore `2100*b^2 = 21,` reduce this equation by 21:
`100*b^2 = 1,` or
`(10*b)^2 = 1.`
Then `10*b=1,` `b=1/10=0.1.`
The answer: `f(x) = 2100*(1/10)^x.`
A completely different answer just occurred to me.
Got technology? Maybe a TI-84 or the equivalent?
Just run an exponential regression! Here's the keystrokes for a TI84. Will probably be similar for other graphing calculators but I will only specifically address TI's.
STAT / EDIT / 1:Edit...
Fill in the x's under L1 and the y's under L2.
STAT / CALC / 0:ExpReg (or scroll down until you see ExpReg)
Confirm the default values (that the Xlist and Ylist are indeed L1 and L2), ignore the rest, hit enter until you get to Calculate.
It will display the values of a and b, and conveniently state the formula for you as well (which is y = a * b ^ x). Plug in the values of a and b and you're all set.
I will start with an intuitive, "common sense" approach, then talk about it in "mathy" terms.
First, the formula for an exponential function like this is y = a(b)^x. The "a" is your initial value, or y-intercept. It's the value of the function when x = 0. The "b" is your common ratio, or what you multiply by every time to get from one term to the next.
Looking at your two points, the first thing that catches my eye is that you are given the y-intercept. Bonus! The value of a is 2100, because the ordered pair (0, 2100) happens to be the y-intercept.
The next thing I look for is the common ratio, or multiplier. In two steps (from x = 0, to x = 1, to x = 2), the value of the function goes from 2100 to 21. A little mental math tells me that I could divide 2100 by 10, twice, to get to 21. More formally speaking… I am multiplying 2100 by 1/10 and by 1/10 again. So the common ratio, or the value of b, is 1/10. (I will address this in more detail in a minute.)
Knowing that the value of a is 2100 and the value of b is 1/10, that gives me a formula of y = 2100(1/10)^x, or y = 2100(0.1)^x, depending on whether your instructor prefers fractions or decimals.
And the “mathy” details…
To find the value of b, you would need to find the nth root of the ratio of the y values, where n is the difference of your x values. Huh? That means you need to divide the 2nd y value by the first y value, or in this case, 21/2100. That’s 0.01. Then you need to take the square root (the second root) of that ratio, because the difference between the x values is 2. (If you’d been given ordered pairs with x values of 2 and 5, you’d find the cube or third root because 5-2=3.) The square root of 0.01 is 0.1. That’s the value of b. Then, you still need the value of a. If you have (0, __ ), or the y-intercept, then you already have a. In this problem, you did, so stop there and use that as the value of a. IF NOT, then your final step is to write as much of the formula as you can, using one ordered pair (x, y) and the value you found for b. In this case, you could write 21 = a(0.1)^2, filling in b (which was 0.1) and the second ordered pair (2, 21). I’ll leave that minor piece of algebra to you, but that gives 2100 when you finish solving for a. Then fill in the formula with a and b and you’re all set.