1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
AVprozaik [17]
2 years ago
14

Write a sine and cosine function that models the data in the table. I need steps to both the sine and cosine functions for a, b,

c, and d. I need a good explantion. Thank you!

Mathematics
1 answer:
dangina [55]2 years ago
4 0

Answer(s):

\displaystyle y = -29sin\:(\frac{\pi}{6}x + \frac{\pi}{2}) + 44\frac{1}{2} \\ y = -29cos\:\frac{\pi}{6}x + 44\frac{1}{2}

Step-by-step explanation:

\displaystyle y = Asin(Bx - C) + D \\ \\ Vertical\:Shift \hookrightarrow D \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \\ Amplitude \hookrightarrow |A| \\ \\ Vertical\:Shift \hookrightarrow 44\frac{1}{2} \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \hookrightarrow \boxed{-3} \hookrightarrow \frac{-\frac{\pi}{2}}{\frac{\pi}{6}} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \hookrightarrow \boxed{12} \hookrightarrow \frac{2}{\frac{\pi}{6}}\pi \\ Amplitude \hookrightarrow 29

<em>OR</em>

\displaystyle y = Acos(Bx - C) + D \\ \\ Vertical\:Shift \hookrightarrow D \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \\ Amplitude \hookrightarrow |A| \\ \\ Vertical\:Shift \hookrightarrow 44\frac{1}{2} \\ Horisontal\:[Phase]\:Shift \hookrightarrow 0 \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \hookrightarrow \boxed{12} \hookrightarrow \frac{2}{\frac{\pi}{6}}\pi \\ Amplitude \hookrightarrow 29

You will need the above information to help you interpret the graph. First off, keep in mind that although this looks EXACTLY like the cosine graph, if you plan on writing your equation as a function of <em>sine</em>, then there WILL be a horisontal shift, meaning that a C-term will be involved. As you can see, the centre photograph displays the trigonometric graph of \displaystyle y = -29sin\:\frac{\pi}{6}x + 44\frac{1}{2},in which you need to replase "cosine" with "sine", then figure out the appropriate C-term that will make the graph horisontally shift and map onto the <em>cosine</em> graph [photograph on the left], accourding to the <u>horisontal shift formula</u> above. Also keep in mind that the −C gives you the OPPOCITE TERMS OF WHAT THEY <em>REALLY</em> ARE, so you must be careful with your calculations. So, between the two photographs, we can tell that the <em>sine</em> graph [centre photograph] is shifted \displaystyle 3\:unitsto the right, which means that in order to match the <em>co</em><em>sine</em> graph [photograph on the left], we need to shift the graph BACKWARD \displaystyle 3\:units,which means the C-term will be negative, and by perfourming your calculations, you will arrive at \displaystyle \boxed{3} = \frac{-\frac{\pi}{2}}{\frac{\pi}{6}}.So, the sine graph of the cosine graph, accourding to the horisontal shift, is \displaystyle y = -29sin\:(\frac{\pi}{6}x + \frac{\pi}{2}) + 44\frac{1}{2}.Now, with all that being said, in this case, sinse you ONLY have a graph to wourk with, you MUST figure the period out by using wavelengths. So, looking at where the graph WILL hit \displaystyle [12, 15\frac{1}{2}],from there to the y-intercept of \displaystyle [0, 15\frac{1}{2}],they are obviously \displaystyle 12\:unitsapart, telling you that the period of the graph is \displaystyle 12.Now, the amplitude is obvious to figure out because it is the A-term, but of cource, if you want to be certain it is the amplitude, look at the graph to see how low and high each crest extends beyond the <em>midline</em>. The midline is the centre of your graph, also known as the vertical shift, which in this case the centre is at \displaystyle y = 44\frac{1}{2},in which each crest is extended <em>twenty-nine </em><em>units</em> beyond the midline, hence, your amplitude. Now, there is one more piese of information you should know -- the cosine graph in the photograph farthest to the right is the OPPOCITE of the cosine graph in the photograph farthest to the left, and the reason for this is because of the <em>negative</em> inserted in front of the amplitude value. Whenever you insert a negative in front of the amplitude value of <em>any</em> trigonometric equation, the whole graph reflects over the<em> midline</em>. Keep this in mind moving forward. Now, with all that being said, no matter how far the graph shifts vertically, the midline will ALWAYS follow.

I am delighted to assist you at any time.

You might be interested in
10 points.............
lions [1.4K]
2nd one is the answer to it.
8 0
3 years ago
Read 2 more answers
An outdoor hockey rink has dimensions of 8m long by 5m wide. The wood
amid [387]

Answer:

First question = 800 cm^3

Second question =120π

Step-by-step explanation:

The formula for volume is base x height x width.

The formula for area of cylinder is 2πrh+2πr^2 but it has no lid so

2πrh+πr^2.

4 0
3 years ago
Which value is a solution of the equation 8y=56?
Korvikt [17]

Answer:

7

Step-by-step explanation:

If you divide 8y / 8  then you have to do the same thing to 56. When you do, y  = 7

3 0
3 years ago
4<br>Order these decimals, starting with the smallest<br>7.96<br>9.61<br>9.534<br>9.5<br>9.53​
Studentka2010 [4]

Answer:

7.96,9.5,9.53,9.534,9.61

7 0
2 years ago
Read 2 more answers
Math
notka56 [123]

Answer:

Step-by-step explanation:

4.5C

6 0
3 years ago
Read 2 more answers
Other questions:
  • Which statements correctly describe the association between the variables A and B?
    9·1 answer
  • You have traveled 150 miles per gallon. how far did you travel per gallon? ​
    12·1 answer
  • The population of a city is expected to increase by 7.5% next year. If p represents the current population, which expression rep
    12·2 answers
  • Wite an expression to represen<br> One minus the product of four and a number
    11·2 answers
  • Solve by factorisation<br> 2x2 + 3x - 20 = 0
    8·1 answer
  • Need help fast plz for my pre calc exam
    5·1 answer
  • HELP ITS A TEST- Question is in the photo
    8·2 answers
  • Can someone help me on this question? Harry and Tim both made New Years resolutions. Harry made 5 more resolutions than Tim. Tog
    13·1 answer
  • A race is 7/10 kilometers long. Maya ran 9 of these races. How far did she run altogether? Write your answer in simplest form.
    7·1 answer
  • A laptop is on sale for 45% off the regular price. If the sale price is $299.75, what was the original price?
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!