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.
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)

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.
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)

AnimeGhoul8863 · Answer

@dude could i get some help please?

AnimeGhoul8863 · Answer

@Hero Could i get some help please?

dude · Answer

Is the equation given

$p(t) = −log10t$ or $p(t) = −log_{10} t$

AnimeGhoul8863 · Answer

No there is no equation given

AnimeGhoul8863 · Answer

^this is a screen shot to show u better

Hero · Answer

@AnimeGhoul8863 if no equation is given then what would you call $p(t) = -\log_{10}t$?

AnimeGhoul8863 · Answer

ummmmmmm imma say Logarithmic? but im probably wrong i say this just because thats what the lesson was called

AnimeGhoul8863 · Answer

im really confused on this stuff

Hero · Answer

What they want you to do is: 
1. Locate the point on your graph where $p(t) = 0$ and $p(t) = 1$. In other words, you have to find the value of $t$ that makes $p(t) = 0$ and $p(t) =1$ You can easily find these points using your graphing calculator.

2. Next they want you to find the value of $p(t)$ when $t = 0.50$. In other words you need to evaluate $p(0.50)$. It will be a value between $0$ and $1$.

3. Now, you'll need to convert the log equation to an exponential equation. Keep in mind that $y = log_b(x)$ is equivalent to $b^y= x$

4. For the last part you'll need to figure out which transformation gives an output for $p$ when $t = 0$.

AnimeGhoul8863 · Answer

1. Locate the point on your graph where p(t)=0 and p(t)=1. In other words, you have to find the value of t that makes p(t)=0 and p(t)=1 You can easily find these points using your graphing calculator.

How do i do this on a graphing calculator how can u find points on a graph with no equation do i just put what they give me and put it in the p(t) = −log10t.?

AnimeGhoul8863 · Answer

i put p(t) = −log10t. in the desmos graphing calculator and the points it says is (0.1, 0)

Hero · Answer

Put $p(t) = -\log(t)$

Hero · Answer

Avoid using the 10. It will only confuse you

AnimeGhoul8863 · Answer

ok

Hero · Answer

The 10 is supposed to be the base of the log but it is not necessary to use in this case. When you use log, the default base is already 10.

AnimeGhoul8863 · Answer

ok

the point i got was (1,0)

Hero · Answer

Very good

Hero · Answer

Anyways, I have to get back to work. Post all your steps and I'll check them later.

AnimeGhoul8863 · Answer

ok....ill try my best but im still confused on what to do

Hero · Answer

What are you confused on?

AnimeGhoul8863 · Answer

everything tbh

AnimeGhoul8863 · Answer

1. Locate the point on your graph where p(t)=0 and p(t)=1. In other words, you have to find the value of t that makes p(t)=0 and p(t)=1 You can easily find these points using your graphing calculator.

if the point is (1,0) is that the value of t

Hero · Answer

The format for points is $(t, p(t))$

AnimeGhoul8863 · Answer

2. Next they want you to find the value of p(t) when t=0.50. In other words you need to evaluate p(0.50). It will be a value between 0 and 1.

if the value of t=(1,0) how is it 0.50 how can u evaluate p(0.50) if its already inbetween 0 and 1

AnimeGhoul8863 · Answer

what

Hero · Answer

Think of steps 1 and 2 as two completely separate tasks. Avoid mixing them up

Hero · Answer

The format for points is like $(x,y)$ except in this case $x = t$ and $y = p(t)$

AnimeGhoul8863 · Answer

ok so if p(t)=0 and p(t)=1 how does t make them (1,0)

AnimeGhoul8863 · Answer

i still dont know

Hero · Answer

Well, yeah, you're clearly confused. I don't really have time to explain it all. It's one thing to be confused about the question itself. It's another thing to also be confused about function notation and points. You're confused about both, which makes explaining more complicated and more confusing for you.

AnimeGhoul8863 · Answer

PART A:

Hero · Answer

Where's the other point? I know desmos gave you that point you already found.

AnimeGhoul8863 · Answer

other point? it doesnt ask for another point

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.

AnimeGhoul8863 · Answer

it ask for a picture of the point on the graph i dont see where there would be a second point

AnimeGhoul8863 · Answer

ughh let me try to find a second point even tho i dont see where one can be

AnimeGhoul8863 · Answer

attachment

AnimeGhoul8863 · Answer

?

AnimeGhoul8863 · Answer

so if Part A: is this

AnimeGhoul8863 · Answer

lets go to part b

AnimeGhoul8863 · Answer

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

AnimeGhoul8863 · Answer

Raising the Hydronium ions to 0.50 would raise the pH levels to 0.301. Once you found this you then can convert the Logarithmic function and in doing this we isolate the x and put it on one side and then take the 10 and put it on the other side creating -10y=x

AnimeGhoul8863 · Answer

i think this is what we do for this part but i could be wrong but i think its right

Hero · Answer

Where is the equation that you graphed? You did not show that in your screenshot. I know how desmos works so why are you hiding that?

AnimeGhoul8863 · Answer

tbh i put it to the side so u could see the graph one second ill get u the other one

AnimeGhoul8863 · Answer

attachment

AnimeGhoul8863 · Answer

i think this is it....yeah it is

Hero · Answer

Okay, great. Your points are correct for part A

AnimeGhoul8863 · Answer

YAY!

AnimeGhoul8863 · Answer

and part b i did the best on

Hero · Answer

As you stated earlier $p(0.50) = 0.30$ which is also correct. Very good. I'm almost suspicious that someone helped you with this.

Hero · Answer

BTW, Do you know to use carets when writing exponents in your equations?

Hero · Answer

Because -10y=x is not correct as written.

AnimeGhoul8863 · Answer

carets no i dont think i have cause ive never heard it i dont think

AnimeGhoul8863 · Answer

let me look it up to see if i have

AnimeGhoul8863 · Answer

is carets where you do x^2

AnimeGhoul8863 · Answer

i see what u mean i think its suppose to be -10^y=x

AnimeGhoul8863 · Answer

is that better

Hero · Answer

It's better

AnimeGhoul8863 · Answer

Ok now part C

AnimeGhoul8863 · Answer

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)

Hero · Answer

Remember to use just $p(t) = -\log(t)$ to express the function. What you copied and pasted above is not correct as written and imprinting the wrong equations into your mind will only confuse and frustrate you if you memorize the wrong ones.

AnimeGhoul8863 · Answer

huh?

AnimeGhoul8863 · Answer

oooo

AnimeGhoul8863 · Answer

The pool company developed new chemicals that transform the pH scale. Using the pH function p(t) = −log(t) 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)

AnimeGhoul8863 · Answer

better

Hero · Answer

Very good.

AnimeGhoul8863 · Answer

so how do we do this

for p(t) + 1 do we do ---> p(t) = −log(t) +1

AnimeGhoul8863 · Answer

o-o ultri why yew stalking

Hero · Answer

Here's what to do. Graph p(t) + 1 on desmos, then evaluate p(0). If you get an output, then you know the transformation produces a y-intercept. If not, try a different transformation.

AnimeGhoul8863 · Answer

evaluate p(0) how can u evaluate on demos?

AnimeGhoul8863 · Answer

*Desmos

AnimeGhoul8863 · Answer

@Hero e.e

Hero · Answer

First graph the function $p(t) + 1$ on the first line then evaluate $p(0)$ on the next line.

AnimeGhoul8863 · Answer

like this ???

Hero · Answer

You want to graph the expression equivalent to $p(t) + 1$

Hero · Answer

You were on the right track earlier

AnimeGhoul8863 · Answer

sorry i got confused let me look up

AnimeGhoul8863 · Answer

im still confused

Hero · Answer

Why? As I said, you were on the right track earlier.

AnimeGhoul8863 · Answer

ik but i dont know how to use desmos what do u mean by "First graph the function p(t)+1 on the first line then evaluate p(0) on the next line. and You want to graph the expression equivalent to p(t)+1"

AnimeGhoul8863 · Answer

i put it on the second line but nothing happened im confused ik i was fine earlier but i done know what to do

Hero · Answer

What is the expression for $p(t)$?

AnimeGhoul8863 · Answer

expression of p(t)?

AnimeGhoul8863 · Answer

ummmm

AnimeGhoul8863 · Answer

p(t) is y is that what ur asking

woolyfrog · Answer

oh no do I have to do this too

Hero · Answer

@AnimeGhoul8863, You already know the expression for $p(t)$. It includes the $\log$ function that was given.

AnimeGhoul8863 · Answer

OHHHH  ok ....but what does that have to do with p(t)+1 and evaluating it with p(0)

AnimeGhoul8863 · Answer

i still dont know how to do that on desmos

woolyfrog · Answer

never mind I think I did this already I'm in segment 2

AnimeGhoul8863 · Answer

Wooly this is segment 2

Hero · Answer

@AnimeGhoul8863 you know that $p(t) = -\log(t)$. So what happens if you add 1 to both sides?

AnimeGhoul8863 · Answer

like p(t)=-log(t) +1

Hero · Answer

Did you add 1 to BOTH sides?

AnimeGhoul8863 · Answer

p(t)= +1-log(t)+1

AnimeGhoul8863 · Answer

?

Hero · Answer

The other +1 goes on the other side of the equal sign.

Hero · Answer

You added 1's to the same side, not both sides.

AnimeGhoul8863 · Answer

ok i get it now hold on i think

AnimeGhoul8863 · Answer

it still isnt working desmos says it only does x and y

Hero · Answer

Remember, though you still have not posted the function here after adding 1 to both sides.

AnimeGhoul8863 · Answer

what

Hero · Answer

$\color{#0cbb34}{\text{Originally Posted by}}$ @Hero
@AnimeGhoul8863 you know that $p(t) = -\log(t)$. So what happens if you add 1 to both sides?
$\color{#0cbb34}{\text{End of Quote}}$

Hero · Answer

You never posted it correctly here.

AnimeGhoul8863 · Answer

p(t)+1=-log(t)+1

Hero · Answer

Very good. Now, on Desmos, post it as $q(t) = -\log(t) + 1$

Hero · Answer

Then evaluate $q(0)$.

AnimeGhoul8863 · Answer

wait what......how did p(t)+1 turn into q(t) what was the point in adding the +1 to that side if we arnt going to use it

AnimeGhoul8863 · Answer

but ok....

Hero · Answer

There's a procedure to this called "Understanding The process".

Hero · Answer

Every step must be understood before moving on to the next step.

Hero · Answer

We could have used $p(t) + 1$ but Desmos refused to accept it.

Hero · Answer

Using Geogebra is far better for tasks like these

AnimeGhoul8863 · Answer

ok

Hero · Answer

Also it is important to realize the expression that is equivalent to $p(t) + 1$. That is why I asked you to do it that way.

Hero · Answer

We know $p(t) = -\log(t)$ After adding 1 to both sides, it follows that $p(t) + 1 = -\log(t) + 1$

AnimeGhoul8863 · Answer

now how do i evaluate on desmos

Hero · Answer

simply type q(0) in the next box after graphing q(t)

Hero · Answer

Screenshot here to make sure that was done correctly.

AnimeGhoul8863 · Answer

attachment

AnimeGhoul8863 · Answer

it wont work

Hero · Answer

No, it worked. q(0) is undefined. What you were supposed to learn here was that if $p(0)$ is undefined then $p(0) + 1$ will also be undefined as adding 1 to both sides does not change the output.

Hero · Answer

So we must try a different transformation.

AnimeGhoul8863 · Answer

ok so we do this one p(t + 1) do we do p(t + 1)=-log(t +1)

Hero · Answer

Correct. Graph it as r(t)

Hero · Answer

Then evaluate r(0).

AnimeGhoul8863 · Answer

so

r(t+1)=-log(t+1)

Hero · Answer

We have to graph the expression on the right as r(t)

Hero · Answer

so r(t) = -log(t + 1)

AnimeGhoul8863 · Answer

nothing happened

Hero · Answer

Show me

AnimeGhoul8863 · Answer

attachment

Hero · Answer

BTW, you forgot the negative and you did get an output for r(0).

Hero · Answer

Remember, 0 is an integer.

AnimeGhoul8863 · Answer

attachment

Hero · Answer

any number you get for an output even if 0, is still an output as 0 is indeed a number.

AnimeGhoul8863 · Answer

but nothing moved like the last one

AnimeGhoul8863 · Answer

but if 0 it is then ok

Hero · Answer

What do you mean "nothing moved". The graph has been translated one unit to the right as it is supposed to.

Hero · Answer

Do you not see that?

AnimeGhoul8863 · Answer

nothing moved but now i see on the corner instead of undefined it says 0

Hero · Answer

Plot points (1,0) and (0,0) to help you understand that p(t + 1) is the graph of p(t) shifted one unit to the LEFT (not right)

Hero · Answer

Other than that, I don't know what you're referring to when you say "nothing moved". The graph of p(t) is definitely shifted.

AnimeGhoul8863 · Answer

nvm

Hero · Answer

What are you thinking is supposed to move besides the graph of p(t) one unit to the left?

AnimeGhoul8863 · Answer

i thought there was a point we are suppose to have this graph acted the same way as the last one thats why i said it didnt move

AnimeGhoul8863 · Answer

but anyway lets continue i need to get this finished

Hero · Answer

Why would it act the same way when it is a different transformation?

AnimeGhoul8863 · Answer

because i thought it was wrong just like the first one

Hero · Answer

We found our y-intercept which is the point (0,0). We're done with this step.

AnimeGhoul8863 · Answer

we are?

Hero · Answer

y-intercept is the point where the graph hits the y-axis.

Hero · Answer

And p(t + 1) hits the y-axis at (0,0). Do you not see this? You do know where the y-axis is or what it is rather, right?

AnimeGhoul8863 · Answer

i see it i got that part and yes i know where the y axis is its horizontal while x is vertical

Hero · Answer

Okay, I figured you did. Just checkin'.

AnimeGhoul8863 · Answer

Okie cx sorry if i seem on edge math stresses me out more than anything

Hero · Answer

but wait, you got that backwards actually. y is vertical, x is horizontal.

AnimeGhoul8863 · Answer

it is?

AnimeGhoul8863 · Answer

aww welp that took my moment away XD

Hero · Answer

Yes, Horizontal means left to right direction. Vertical means up and down direction.

Hero · Answer

I have to finish my work so finish up and I'll check what you did later.

AnimeGhoul8863 · Answer

Thanks Hero <3

i have to submit this right now but i believe its correct you helped me lots which means alot and sorry for getting confused math is just not my greatest subject

Hero · Answer

You're welcome. Happy to help.

1 attachment

1 attachment

1 attachment

1 attachment

1 attachment

1 attachment

1 attachment

1 attachment

1 attachment