Exponential Function

Question

Write the exponential function that corresponds to this set of values: {(1, 90), (2, 81), (3, 72.9), (4, 65.61)}

Answer 1

An exponential function is one of the form

y = a * b^x

We know the following:

90 = a * b¹

81 = a * b²

72.9 = a * b³

65.61 = a * b⁴

By dividing the second equation by the first, we can easily see that

81 / 90 = (a * b²) / (a * b¹)

And simplifying, we get 0.9 = b.

Plugging back in to any of the four equations, we see that.

90 = a * 0.9

And thus a = 100.

Hence the function is y = 100 * 0.9^x.

Hope that helps!

Answer 2

Write the exponential function that corresponds to this set of values: {(1, 90), (2, 81), (3, 72.9), (4, 65.61)}

exponential function look like this: y=b^x

^{1st: 90~~~~~b^1}

^{2nd:81~~~~~b^2}

^{3rd: 72.9~~~~b^3}

^{4th: 65.61~~~b^4}

^{I noticed that I can find some kind of relationship using 2nd and 4th; when I square 2nd one, I get 81^2~~~~~~~~b^4.}

^{Now, look at 81^2 and 65.61(4th one). 81^2=6561, therefore, 6561=65.61 x 100. This means in the function I am looking for 100 is involved somewhere. Basic form of exponential function look like:} y=b^x

I was given four y values and four x values, then 100 must go to b, so let's just set y=100(b^^x). And think about it. I go back and look at 1st: 90~~~~~b^1. And I substitute y=90 and x=1 into y=100(b^^x) to get, 90=100(b^¹). when I divide by 100 both side, I get 0.9=b^¹ . Now, I got answer, 0.9=b.

At this point, go back "let's just set y=100(b^^x)." substitute b=0.9, I get y=100 x (0.9)^^x.

^{Note: I used 2nd, 4th, and 1st to get} y=100 x (0.9)^^x.

^{I did not use 3rd: 72.9~~~~b^3. Test see if 3rd one work. Is it true 72.9=100 x (0.9)^3 ? Yes.}

Answer is y=100 x (0.9)^^x .

Tip: In a problem involves function, try to find relationship. The process start from using simple numbers.

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