Can anyone help explain how to do this? I'm behind a little and super confused.

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.

1. 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.
2. Use your graph to find the ideal pH level if the amount of hydronium ions is raised to 0.50.
3. 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.
4. f(x) + 1
5. f(x + 1)

I don't really understand what I'm supposed to do, so any help will be greatly appreciated. Thank you in advance!

Answer 1

All you have to do is plot a graph of f(x) = -log₁₀(x) for part 1. The f(0) is asymptotic to the y-axis and f(1) = 0.

2) The problem only wants f(.5) which is -log₁₀(1/2). You can either look at your graph or just use your calculator to get around 0.301.

3) The problem wants you to do the two function transformations.

Remember the transformation rule g(x) = Af(x+B) + C. A = amplitude change. B = the units to shift the curve left and C = the vertical change of the function.

For f(x)+1, just raise the whole function curve up one unit. Or g(x) = 1-log₁₀(x)

For f(x+1), shift the curve left one unit. Or g(x) = -log₁₀(x+1).

If you need more help, we could schedule a session.