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 0.35. 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 0.35. 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. (You must have all 3 graphs and an explanation of each transformation).

p(t) + 2
p(t + 2)
−2 • p(t)

Analyze the two functions. Answer the following reflection questions in complete sentences.
What is the domain and range of the rainbow?

Explain what the domain and range represent in context.

Do all of the values make sense in this situation? Why or why not?

What are the x- and y-intercepts of the rainbow?

Explain what each intercept represents in context.

Is the linear function you created with your table positive or negative? Explain.

What are the solutions or solution to the system of equations created? Explain what it or they represent.

Create your own piecewise function with at least two functions. Explain, using complete sentences, the steps for graphing the function. Graph the function by hand or using a graphing software. Remember to restrict your domains so your function will pass the vertical line test.