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enter image description hereThe question is a straight forward 7th grade question with no exposure to logs. Given some parent graph shown in red, (which is easy for me to see is $y=2^x$, what is the transformed graph shown in blue, given that the transformed graph looks like (see included) and has the following 2 ordered pairs $(2,25), (-1,0.2)$. It's easy for me to see that the new function is $y=5^x$. However, this represents a change in the "b" value of $y=ab^x$ and I know that for this transformation it should be a horizontal compression or stretch NOT a change in the "b" value, but rather a change in the coefficient in front of the $x$ term.

How do I determine what the transformation is? These values are easy to see, but I want to understand the process so that if I had more complex problems I could see what the process is.

My thought is that given the parent graph of $y=ab^x$, the transformation would represent a change in the coefficient in front of the $x$ value. Specifically, there should be in the transformed graph a term $y=ab^\left(cx\right)$, where I need to determine the value of c.

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  • $\begingroup$ suppose $c = \log_b 5$. $\endgroup$ – John Joy Feb 23 at 7:12
  • $\begingroup$ absolutely. That is the answer. However, as I said in the question, I'm trying to explain this concept to daughter without using logs, since she has not been introduced to logs in 8th grade. $\endgroup$ – user163862 Feb 23 at 19:03
  • $\begingroup$ The instructor is saying that the transformation of the parent function is $y=5^x$. But, my understanding of transformation of parent functions is that you cannot change the parent function "b" value. $\endgroup$ – user163862 Feb 23 at 19:04
  • $\begingroup$ Perhaps and "existence" explanation then? we know that $2^2 = 4$ and $2^3 = 9$. If you assume that the function is increasing and continuous, then one can assume that there is some value $c$ such that $2^c = 5$. That would give us $y = 1\cdot 2^{cx} = (2^c)^x=5^x$ for some value $c|2<c<3$ $\endgroup$ – John Joy Feb 23 at 21:58
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I now see the answer to this, without using logs! Since the mapping for the translation of the parent graph $y=2^x$ to $y=2^\left(cx\right)$ is $(x,y)$ maps to $((1/c)x,y)$ then I only need to go to the graph and find a constant $y$ value and compare the ratio of the $x$ values. For example, for the parent graph there is a point $(1,2).$ If I look along the transformed graph to find the value of the $x$ that corresponds to a $y$ value of $2$, I see that that value is approximately $0.432$. Thus $1/c=0.432$ (approximately, within the constraints of reading the graph) and so $c= 2.32$ (again, approximately). I know that the real answer is found by using logs, but this method gives you that answer without using logs.

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