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I have a question about a Khan Academy problem for determining an exponential function formula based on information from a graph.

The problem: $h(x)= a \times r^x$

We are given two points. (-2, 4.5) and (0,8)

I'm having a problem with one of the steps in the hints section. It's not all clear to me what they did to get the value. I'll explain:

Firstly we know that the 8 in (0,8) is going to be our "a" in the formula.

The problem I'm having is when we plug the next set of points into the equation.

$4.5 = 8 \times r^{-2}$

$\frac{9}{2} = \frac{8}{r^2} $

I understand how the right side gets converted to $8/r^2$ but I don't understand why 4.5 gets multiplied by 2 when all we did on the right side was convert $r^-2$ to $1/r^2$ and multiply by 8. There's no details associated with the any of the 4 steps that they work out and I've been racking my brain trying to figure out a solution with no luck. Any help is greatly appreciated.

Kanwaljit Singh
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BoutTreeFiddy
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1 Answers1

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As $4.5 = \frac{9}{2}$

So,

$\frac{9}{2} = \frac{8}{r^2}$

Cross multiply terms.

$9 * r^2 = 8 * 2$

$9 * r^2 = 16$

$r^2 = \frac{16}{9}$

$r = \sqrt{\frac{16}{9}}$

$r = \frac{4}{3}$

Kanwaljit Singh
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  • Why does 4.5 get multiplied by 2 on the left side when on the right side we're converting the negative exponent and multiplying 8? I don't understand where the 2 comes from. I understand everything else. – BoutTreeFiddy Dec 16 '16 at 16:28
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    2 is not multiplying it is dividing 9. When 9 divided by 2 equals to 4.5 So we replaced 4.5 by 9/2. – Kanwaljit Singh Dec 16 '16 at 16:35
  • Easiest way if you remove decimal in 4.5 you will get 45/10 divide them by 5. You get 9/2. So either use this method or directly replacing method used in above comment. – Kanwaljit Singh Dec 16 '16 at 16:37
  • If still any problem let me know. Happy to help you. – Kanwaljit Singh Dec 16 '16 at 16:37
  • I don't understand why in the same step that the negative exponent for "r" get converted and multiplied by 8, 4.5 magically gets multiplied and becomes 9/2 along side it. I'm failing to see where the 2 that multiplies 4.5 into 9/2 is coming from. Is it just done solely out of convenience to solve the rest of the problem without the 2 that's multiplying 4.5 originating from some where in the problem? This whole self taught thing has its cons if you haven't noticed. This would be easier if I had a teacher. Thank your for your patience and help! I truly do appreciate it. – BoutTreeFiddy Dec 16 '16 at 16:44
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    Oh my goodness I feel dumb haha! I've been spending way too much time cramming knowledge that I'm over analyzing incredibly simple steps. 4.5 is the same as 4 1/2 and converts to an improper fraction...9/2. Thank you for taking the time to answer my question, as dumb as it was. :| – BoutTreeFiddy Dec 16 '16 at 16:57
  • Nice you got it. Any other doubt? – Kanwaljit Singh Dec 16 '16 at 16:58
  • I think with $r^{-2}$. When we remove negative sign from power. Just simply do the reciprocal of that term. Like $r^{-2}$ become $1/r^2$ – Kanwaljit Singh Dec 16 '16 at 17:04
  • And if all doubts cleared then you can tick my answer and also upvote it if you like. – Kanwaljit Singh Dec 16 '16 at 17:14