If you ever swam in a pool and your eyes began to sting and turn red, you felt the effects of an incorrect pH level. pH measures the concentration of hydronium ions and can be modeled by the function p(t) = −log10t. The variable t represents the amount of hydronium ions; p(t) gives the resulting pH level.

Water at 25 degrees Celsius has a pH of 7. Anything that has a pH less than 7 is called acidic, a pH above 7 is basic, or alkaline. Seawater has a pH just more than 8, whereas lemonade has a pH of approximately 3.

Create a graph of the pH function either by hand or using technology. Locate on your graph where the pH value is 0 and where it is 1. You may need to zoom in on your graph.
The pool maintenance man forgot to bring his logarithmic charts, and he needs to raise the amount of hydronium ions, t, in the pool to 0.50. To do this, he can use the graph you created. Use your graph to find the pH level if the amount of hydronium ions is raised to 0.50. Then, convert the logarithmic function into an exponential function using y for the pH.
The pool company developed new chemicals that transform the pH scale. Using the pH function p(t) = −log10t as the parent function, explain which transformation results in a y-intercept and why. You may graph by hand or using technology. Use complete sentences and show all translations on your graph.
p(t) + 1
p(t + 1)
−1 • p(t)

For a better understanding of the explanation provided here, please go through the graphs in the attached files. The first file contains the explanations to the first part of the question and the second file contains the three graphs required to explain the second part (the y intercept part) of the question.

It has been given to us that the pH concentration equation is:

[tex] p(t)=-log_{10}t [/tex]

where, The variable t represents the amount of hydronium ions and p(t) gives the resulting pH level.

As can be seen from the first graph, pH is 1 when the amount of hydronium ions, t is 0.1. This is as expected, because, from the concept of logarithms we know that:

[tex] log_{10}(0.1)=log_{10}(\frac{1}{10})=log_{10}(10^{-1})=-1\times log_{10}10=-1\times 1=-1 [/tex]

Thus, [tex] p(0.1)=-1\times-1=1 [/tex]

Again, the value of pH, p(t) is 0 when the concentration of the Hydronium ions, t is 1. This, again, is as expected, because, from the concept of logarithms we know that [tex] log(1)=0 [/tex].

If the pool maintenance man uses our graph and needs to find the the pH level if the amount of hydronium ions is raised to 0.50, then to do so, the pool maintenance man must draw a vertical line from t=0.5 and check the x coordinate of the point where this vertical line intersects the p(t) graph. As we can see from the graph this roughly comes out to be the horizontal line y=0.3.

Let us now move on to the second part of the question. For a better understanding of this part please go through the second graph which has been attached. This graph contains all translations on the original graph and the explanation to each one of them is provided below:

Let us start with the last one, -p(t). We know that, since, [tex] p(t)=-log_{10}(t) [/tex], therefore, [tex] -p(t)=log_{10}(t) [/tex], simply a vertical "flip" or reflection of the original graph. And because the original graph did not have a y intercept, this graph too will not have a y intercept.

Let us now move on to the first one. Here, [tex] p(t)+1=-log(t)+1 [/tex]. This graph will simply move the original graph vertically up by 1 unit and that will not have any bearing on the y intercept and thus, the graph of p(t)+1 will again, not have a y intercept.

Finally, let us move on to the graph of the equation p(t+1). As can be seen, this graph has a y intercept at the origin when t=0. This is because:

[tex] p(t+1)=log_{10}(t+1) [/tex]

and when t=0, [tex] p(t+1)=p(0+1)=log_{10}(0+1) [/tex]

[tex] p(1)=log_{10}(1)=0 [/tex]

Thus, when t=0, p(t+1)=0 too and thus we have a y intercept.

Therefore, out of the three transformations, only the middle transformation results in a y-intercept, the explanation for which has been provided.