Question

Have you ever been swimming in a pool, and your eyes were red when you get out? This means the chemicals in the pool were not adjusted correctly! More specifically, the measurement of the pool's pH was not quite right.
pH is a measurement of hydronium ions and can be modeled by the function f(x) = −log10x. The variable x represents the amount of hydronium ions, and f(x) gives the pH level.

Different liquids have ideal pH values that are different. Water at 25 degrees Celsius has a pH of 7. Anything that has a pH less than 7 is called acidic, and 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.
Use your graph to find the ideal pH level if the amount of hydronium ions is raised to 0.50.
The pool company developed new chemicals that transform the pH scale. Using the pH function f(x) = −log10x as the parent function, explain which transformation results in a y-intercept and why. You may graph by hand or using technology. Be sure to label each transformation on the graph.
f(x) + 1
f(x + 1)

Answer 1

The graph of the pH function can be created by plotting the output of

input values of the hydronium ions, x, in the range 0 < x ≤ 1.

Correct responses:

First part:

Please find attached the graph of the pH function

From the attached graph of the pH function, we have;

• The point where the pH value is 0 is $\underline{(1, \, 0)}}$

• The point where the pH value is 1  is $\underline{(0.1, \, 1)}$

• The pH level if the hydronium ion is raised to 0.50 is approximately 0.3

Second part

• Please find attached the graph of the transformations, f(x) + 1, and f(x + 1) of the parent log function.

• The transformation that results in a y-intercept is $\underline{f(x, \, 1)}$, because, at x = 0, f(x + 1) = 0.

Which method can be used to find the pH values using a graph?

First part

The pH function is; f(x) = -log₁₀x

Please find attached the graph of the pH function created with MS Excel

Where;

x = Amount of hydronium ions

f(x) = pH level

When the pH is 0, we have;

-log₁₀x = 0

Which gives;

x = 10⁰ = 1

• When the pH value is 0, the point on the graph is $\underline{(0, 1)}$

When the pH value is 1, we have;

-log₁₀x = 1

log₁₀x = -1

x = 10⁻¹ = 0.1

• The point on the graph where the pH value is 1 is $\underline{(0.1, \, 1)}$

• From the graph, the ideal pH level if the amount of hydronium ions is raised to 0.50 is pH = 0.301029996 ≈ 0.30

Second Part

The given transformations are;

f(x) + 1, and f(x + 1)

• The graph of the transformations created with MS Excel is attached

At the y-intercept, x = 0

From the transformation, f(x) + 1, we have;

At x = 0, the transformation = -log₁₀(0) + 1

log₁₀(0) = Does not exist (DNE)

Therefore;

f(x) + 1 = -log₁₀(0) + 1 = DNE

The transformation, f(x + 1), at x = 0, gives;

f(0 + 1) = -log₁₀(0 + 1) = -log₁₀(1) = 0

Therefore;

The function f(x + 1) = -log₁₀(x + 1) is defined at x = 0, and therefore, has a y-intercept

• The transformation that has a y-intercept is $\underline{f(x + 1)}$

Learn more about the pH values here:

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