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.

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)

If we have the function [tex]p(t) = -log_{10}(t)[/tex] as the parent function then we must find out which of the transformations shown must be performed to the function so that it has an intersection on the y axis.

The main function [tex]p(t) = -log_{10}(t)[/tex] because if we do t = 0 then p(t) is not defined because the log(0) is not defined.

Then, if we transform the function by doing [tex]y = p(t) +1[/tex] then we will obtain:

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

This transformation moves the parent function 1 unit up. Therefore, if the original function had no interception in y, then it will not have an intercept either.

Then, if we transform the function by doing [tex]y = p(t + 1)[/tex] then we will get:

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

This function moves the parent function one unit to the left on the x axis. Therefore it will have an intercept in p(t) = 0

The transformation [tex]-1*p (t)[/tex] only reflects the function p(t) on the x axis. But if the main function has no intercept on the y-axis then the reflected function will also have no intercept on the y-axis

Therefore the correct option is the second one. The transformation that will cause p(t) to have an intercept with the y = p(t) axis is [tex]p(t) =-log_{10}(t+1)[/tex]

Observe the attached image