Captain K and his crew of math genius' continued their travel through The Land Of Pre-Calculus in search of the Legendary Holy Golden Credit!! When they came across an obstacle in their way of eternal bliss.....A SEA PORT! This sea port was the only way they were able to cross the sea without being swept away by its' tides of sinusoidal functions.
"What do we do now Captain K?" said Craigmyer.
"Look there's a sign near by, let's see what it says" said Melk
"Trying to get across the sea port? It's not going to be as easy as walking across, first you must solve these problems. Then you shall see what happens next..." read the crew.
At this sea port, the depth of the water, h meter, at time, t hours, during a certain day is given by this formula:
h(t) = 1.8sin[2Π (t-4.00)/12.4] + 3.1
a) State the: (i) period (ii) amplitude (iii) phase shift
b) What is the maximum depth of the water? When does it occur?
c) Determine the depth of the water at 5:00 am and at 12:00 noon.
d) Determine one time when the water is 2.25 meters deep.
"WHAT ARE WE SUPPOSE TO DO!?!?!" yells Danny Boy frantically
"No need to worry Bertman is here! We just have to sketch a graph of this sea and we'll get our answers from there!" says Bertman speaking in a voice strangely close to that of Batman's.
"Easier said then done, but I'll give it a try" says Mr. Siwwy.
"What a minute, we don't even need the graph for the first part of the question!" states Kasiaw Cole (get it, Keyshia Cole, the singer?). She continues, "to find the period don't you have to go 2Π/B? But that fraction, that makes it so much harder then usual."
"I got it!" says Vinsanity "you just got to multiple it all out first, then this will give you the answer for the period"
Vinsanity shows his work to the rest of the crew:
2Π X (t-4)/12.4
= 2Π/1 X (t-4)/12.4
= 2Π (t-4)/12.4
= 2Π/12.4 X (t-4)
B = 2Π/period
B = 2Π/12.4
PERIOD = 12.4
"Nice! I got the amplitude and phase shift!" says Dr. Grey-M "By looking at the formula we get the amplitude to be 1.8 and the phase shift to be 4.00."
"Awesome, on to the next questions!" says Danny Boy
NOTE: This is where my story kind of just stops for a bit and I just post the answers to the remaining questions b), c), d). This is taking way longer then I expected and it's already cutting into my sleep time, forget about my studying time =S. Continuing......
b) The maximum depth of the water is 4.9 meters. You get this by taking your sinusoidal axis and adding the amplitude to it:
3.1 + 1.8 = 4.9 meters
To find when it occurs you plug in 4.9 into the formula:
4.9 = 1.8sin[2Π (t-4.00)/12.4] + 3.1
1.8 = 1.8sin[2Π (t-4.00)/12.4]
1 = sin[2Π (t-4.00)/12.4]
Π/2 = [2Π (t-4.00)/12.4]
(12.4)(Π/2) = 2Π (t-4.00)
(12.4)(Π/2) / (2Π) = t-4.00
3.1 + 4.00 = t
7.1 = t
7:06 am = t
c) For each time, you just plug them into the formula to find your depth at that time:
h(5 am) = 1.8sin[2Π (5-4)/12.4] + 3.1
h(5 am) = 3.9735 m
h(12 noon) = 1.8sin[2Π (12-4)/12.4] + 3.1
h(12 noon) = 1.6766 m
NOTE: If you were given a time value of 5:00 pm this would be equal to 17 hours, and that would be the value you plug into the formula.
d) To determine one time when the depth of the water is 2.25 meters deep, we just plug 2.25 meters into the formula:
2.25 = 1.8sin[2Π (t-4)/12.4] + 3.1
Let Θ = [2Π (t-4)/12.4]
2.25 = 1.8sinΘ + 3.1
-0.85 = 1.8sinΘ
-0.4722 = sinΘ
-0.4918 = Θ
-0.4918 = [2Π (t-4)/12.4]
(12.4)(-0.4918) = 2Π (t-4)
(12.4)(-0.4918) / 2Π = t-4
-0.9706 + 4 = t
3.0294 = t
BACK TO THE STORY:
"Wow, that took a long time but we finally figured it out" says Richard S.
Just then they saw a magical bridge appear out of no where allowing them to cross the sea safely and on towards their next challenge while in search of....the Legendary Holy Golden Credit!!
"Hey how did you get here before we did Samus?" says Dino
"Didn't you guys see that perfectly safe bridge right down over there?" Samus points towards the bridge.
"WHY DIDN'T YOU TELL US!!??!!??" the crew screams at Samus.
"You never asked? DUH!" says Samus
PART TWO OF THE STORY: Transformator And His Evil Pre-Test..... NOTE: I'm going to use past colours again, there's too many people in our class and not enough colours =S
So after getting past the sea port and it's puzzling questions, our adventurers ventured onward along the "yellow brick road" not knowing that they were about to encounter the EVIL TRANSFORMATOR DUN DUN DUN!
"Why are we walking on this yellow road anyways?" says KaDeeM AbDul Ali JaBBar confused.
"Captain K told us to, he said this road would lead us to......the Legendary Holy Golden Credit!!" says Tim_MATH_y
"But if it lead us to....the Legendary Holy Golden Credit!! I really wish that'd stop happening everytime we mention....the Legendar Holy Golden Credit!! Sigh, anyways, if it lead us to that, then wouldn't the road be gold instead of yellow? Guys? Guys?" says KaDeeM AbDul Ali JaBBar
He turns around only to see the whole crew staring at the biggest transformation ever seen on the face of The Land Of Pre-Calculus, TRANSFORMATOR!!!
"MUAHAHAHAHA!!!! I'm Transformator and I'm here to phase and vertically shift your functions to doom! Then when I'm done with that, I'll stretch and squash your amplitude until you scream for your MAMA!! MUAHAHAHA!!!" says the hideous Transformator
"Ewww, this thing is so ugly it makes me sick. What are we going to do? Stupid Transformator we just want to get the Golden Credit. Oh my gosh leave us alone." says Jeng-Lo
"Never! You're trespassing on my territory there's nothing you can do now to escape from my parameters A, B, C, and D!" says Transformator
"Alright then, I purpose a challenge to you Transformator. I bet our whole crew together can solve any transformation problem you throw at us. If we win then you let us go." says Sandy softly but bravely.
"And if you lose?" says Transformator
"Then we have to stay here forever allowing you to do whatever you want to us" says Sandy
"Alright then, you have a deal! Here's a Pre-test chalk full of challenging problems, lets see if you can get out of this one Captain K! MUAHAHAHA!" says Transformator in delight.
The crew recieves the test and starts to begin solving the questions:
1. f(x) = 2x² - 3, where x is greater than or equal to 0, then a function g that will have domain and range that are both different from those of function f is:
a) g(x) = f(-x)
b) g(x) = -f(x)
c) g(x) = f-¹(x)
d) g(x) = kf(x), k greater than 0
"Oh! The answer is C)!" says Aichelle the Incredible
"The answer wouldn't be A) because f(-x) will only result in a different domain, and it wouldn't be B) because -f(x) only results in a different range. For C), f inverse of (x) means your x coordinates would be your y coordinates, and your y coordinates would be your x coordinates therefore giving you a different domain and range!" continues Aichelle the Incredible
"Argh, that's correct, next question" says Transformator angered
2. The graph of a function f is a parabola opening upward, with its vertex on the x-axis. The graph of a new function g, where g(x) = 2f(x), will have:
a) the same domain and the same range as f
b) the same domain but a different range than f
c) a different domain but the same range as f
d) a different domain and a different range than f
"Jojo Rocks!" yells Jojo
"The answer has to be A)! Since it's a parabola, going 2f(x) wouldnt affect the range because the parabola would be going up both sides to infinity so you could trash B) and D) already. It can't be C) either because the vertex is on the x-axis, therefore the domain will be left unchanged so the answer has to be A)!" continues Jojo
"ARGH! Right again! Next one, you won't get this!" grumbles Transformator
3. Given the graph of f(x) below, sketch 1/f(x) in the space provided:
"First things first find your invariant points, on this graph there's two. One at (-1,-1) and the other at (1,1). Then find your asymtotes, it just so happens that the asymtotes are the x and y-axis. After that just remember Dr.Suesus' version of math when drawing the actually reciprocal graph in (smallering and biggering). Also remember to have arrows where neccesary and dots that shows an end on the graph where necessary. In this case the graph ends at (-5,-1) but continues on going in the other direction." states Johnny Johnson in what seems like an eternity to explain.
"NO!! RIGHT AGAIN!! THERE'S NO WAY YOU'LL GET THESE LAST TWO!" screams Transformator
4. Given f(x) = cube√3x² -4, write the equation for its inverse f-¹(x).
"They just keep getting easier and easier! I'll show you the work, no sweat." says Bond, Robert Bond
"THIS CAN'T BE HAPPENING!! THIS LAST QUESTION IS IMPOSSIBLE, NOT EVEN I WAS ABLE TO SOLVE IT!!!" yells Transformator, now shaking the ground with his mighty power.
And this is where I'm going to stop for tonight. Sorry guys I put out more then I could chew, but this final question will be up by tomorrow! Good luck to all on the test in the AFTERNOON. Scribe as you all know already is Mark. =D