If you ever swam in a pool and your eyes began to sting and turn red,

Question

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 by 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)

Answer 1

It appears the question is asking you to use the function p(t) = log₁₀(t). This is a really interesting formula to model pH and this is my first time hearing about it! We must find out when our function p(t) will be at 0 and 1.

You can evaluate this by hand or use a graphic calculator (which is easier). I have gone ahead and put this into Desmos. I can see on here that p(t) is 0 at t = 1 and p(t) = 1 at 0.1.

It seems part of this question is missing. If you're able to find it, please respond!

Finally, we see that it is only p(t+1) that creates a y intercept.

The function p(t) does not cross the y axis. This is because in the question, the variable t is the argument. If t were negative, this is basically saying that 10 to the power of something can make a negative number, and there is no number that can make this possible.

So the function p(t) never crosses the y axis, and a vertical reflection will not cause it to cross. Also, a vertical translation upward will not do that either. It is only when we move the whole thing to the left, which is done with horizontal translations (when you add or subtract inside the parentheses) that we can create a y intercept with this function!