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 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.

Recall from 07.02 how to convert a log to an exponential function. The equation y equals b to the power of x is written on the left. The equation x equals log base b of y is written on the right. A double-ended arrow points to both equations. The word exponent is written above the equations with arrows pointing to the x on the left and the x on the right. The word base is written below the equations with arrows pointing to the b on the left and the b on the right.
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

The answer to the final question is
`p(t+1)`.

Explanation:

Attached there are three figures, each one related to one question.

First: In the figure the graph of the pH function is represented, and the GeoGebra software was used to obtain the graph. Notice how the graph of the pH function approaches the y-axis but do not intersects it. In the figure there are, also, two points: the intersects with the x-axis which occurs at (1,0) an the point (0.1,1), where the value of the pH function is 1.

Second: The second figure is a zoom of the previous figure, where the points (1,0) and (0.1,1) are plotted. Moreover, we have the point H with

abscissa 0.32, which is the value where the pH function gives 0.5 approximately.

Third: The graph of the function
`p(t+1)` is plotted. The fact of the y-axis intersects can be checked easily. In order to answer this question we need to recall how the graph of a function
`f(x)` is affected after a translation:

• If we make
  `f(x)+a` we make a translation of
  `a` units in the y-axis. So, the function moves up or down in dependence of the sign of
  `a`

• If we make
  `f(x+b)` we make a translation of
  `-b` units in the x-axis. So, the graph of the function moves to the right if
  `b<0` or it moves to the left if
  `b>0`.

In this case, as we have
`p(t+1)`, we know that the graph of
`p(t)` moves one unit to the left. Then, it intersects the y-axis.