# 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 $c = \log_b 5$. Feb 23 '19 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. Feb 23 '19 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. Feb 23 '19 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 $c|2<c<3$ Feb 23 '19 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.