# Determine transformation of parent graph given new graph and 2 ordered pairs

The 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.

• suppose $c = \log_b 5$. – John Joy Feb 23 at 7:12
• 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. – user163862 Feb 23 at 19:03
• 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. – user163862 Feb 23 at 19:04
• 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 $c|2<c<3$ – John Joy Feb 23 at 21:58

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.