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sergiy2304 [10]
2 years ago
8

PLEASE HELP QUICKLY 25 POINTS

Mathematics
2 answers:
Natalija [7]2 years ago
7 0

Answer:

○ \displaystyle \pi

Step-by-step explanation:

\displaystyle \boxed{y = 3sin\:(2x + \frac{\pi}{2})} \\ 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 0 \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \hookrightarrow \boxed{-\frac{\pi}{4}} \hookrightarrow \frac{-\frac{\pi}{2}}{2} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \hookrightarrow \boxed{\pi} \hookrightarrow \frac{2}{2}\pi \\ Amplitude \hookrightarrow 3

<em>OR</em>

\displaystyle \boxed{y = 3cos\:2x} \\ 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 0 \\ Horisontal\:[Phase]\:Shift \hookrightarrow 0 \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \hookrightarrow \boxed{\pi} \hookrightarrow \frac{2}{2}\pi \\ Amplitude \hookrightarrow 3

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 photograph on the right displays the trigonometric graph of \displaystyle y = 3sin\:2x,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>sine</em> graph [photograph on the left], accourding to the horisontal shift formula 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 [photograph on the right] is shifted \displaystyle \frac{\pi}{4}\:unitto the right, which means that in order to match the <em>cosine</em> graph [photograph on the left], we need to shift the graph BACKWARD \displaystyle \frac{\pi}{4}\:unit,which means the C-term will be negative, and by perfourming your calculations, you will arrive at \displaystyle \boxed{-\frac{\pi}{4}} = \frac{-\frac{\pi}{2}}{2}.So, the sine graph of the cosine graph, accourding to the horisontal shift, is \displaystyle y = 3sin\:(2x + \frac{\pi}{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 [-1\frac{3}{4}\pi, 0],from there to \displaystyle [-\frac{3}{4}\pi, 0],they are obviously \displaystyle \pi\:unitsapart, telling you that the period of the graph is \displaystyle \pi.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 = 0,in which each crest is extended <em>three units</em> beyond the midline, hence, your amplitude. So, no matter how far the graph shifts vertically, the midline will ALWAYS follow.

I am delighted to assist you at any time.

givi [52]2 years ago
6 0

Answer:

\pi

Step-by-step explanation:

The trigonometric function shown on the graph is a <u>continuous function</u>, and so its period is the <em>length </em>of one wave of the function (i.e. the horizontal distance between a peak and the next peak, or a trough and the next trough).

From inspection of the graph, the horizontal distance between one peak and the next of the given function is \pi.  

Therefore, the period of this trigonometric function is \pi

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{\large{\textsf{\textbf{\underline{\underline{Question \: 1 :}}}}}}

\star\:{\underline{\underline{\sf{\purple{Solution:}}}}}

\bullet \sf \:   {(a + b)}^{ab}

<u>Putting value of a as 3 and b as -2, we get</u><u> </u><u>:</u>

\longrightarrow \sf \:   {( 3 +  (- 2))}^{3 \times  - 2}

\longrightarrow \sf \:   {( 3 - 2)}^{3 \times  - 2}

\longrightarrow \sf \:   {( 1)}^{ - 6}

• <u>Using negative Exponents Law</u>

\longrightarrow \sf   \dfrac{1}{ {1}^{6} }

\longrightarrow \sf   \dfrac{1}{ 1 \times 1 \times 1 \times 1 \times 1 \times 1 }

\longrightarrow \sf   \dfrac{1}{  1 }

\longrightarrow \sf   \purple{1}

{\large{\textsf{\textbf{\underline{\underline{Question \: 2 :}}}}}}

\star\:{\underline{\underline{\sf{\red{Solution:}}}}}

\bullet  \sf \:  \dfrac{ {8}^{ - 1} \times   {5}^{3} }{ {2}^{ - 4}}

\longrightarrow  \sf \:  {8}^{ - 1} \times   {5}^{3}  \times  \dfrac{1}{{2}^{ - 4}}

<u>• Using negative Exponents Law</u>

\longrightarrow  \sf \:  {8}^{ - 1} \times   {5}^{3}  \times   {2}^{4}

\longrightarrow  \sf \:  {8}^{ - 1} \times   5 \times 5 \times 5  \times   {2}^{4}

\longrightarrow  \sf \:  {8}^{ - 1} \times 125  \times   {2}^{4}

\longrightarrow  \sf \:  {8}^{ - 1} \times 125  \times   2 \times 2 \times 2 \times 2

<u>• Using negative Exponents Law</u>

\longrightarrow  \sf \:   \dfrac{1}{ \cancel{8}_{4}} \times 125  \times   \cancel{2}_{1} \times 2 \times 2 \times 2

\longrightarrow  \sf \:   \dfrac{1}{ \cancel4_{2}} \times 125  \times   \cancel{2}_{1}  \times 2 \times 2

\longrightarrow  \sf \:   \dfrac{1}{ \cancel2} \times 125  \times   \cancel{2}   \times 2

\longrightarrow  \sf \:    125  \times 2

\longrightarrow  \sf \red{  250}

{\large{\textsf{\textbf{\underline{\underline{Question \: 3 :}}}}}}

\star\:{\underline{\underline{\sf{\green{Solution(1):}}}}}

\bullet \sf  \dfrac{ \sqrt{32} +  \sqrt{48}  }{ \sqrt{8} +  \sqrt{12}  }

\longrightarrow \sf  \dfrac{ \sqrt{4 \times 4 \times 2} +  \sqrt{4 \times 4 \times 3}  }{ \sqrt{2 \times 2 \times 2} +  \sqrt{2 \times 2 \times 3}  }

\longrightarrow \sf  \dfrac{ \sqrt{  {4}^{2}   \times 2} +  \sqrt{ {4}^{2}  \times 3}  }{ \sqrt{ {2}^{2}  \times 2} +  \sqrt{ {2}^{2}  \times 3}  }

\longrightarrow \sf  \dfrac{ 4\sqrt{    2} + 4 \sqrt{  3}  }{ 2\sqrt{  2} +2  \sqrt{  3}  }

\longrightarrow \sf  \dfrac{ \cancel{ 4}_{2}(\sqrt{    2} +  \sqrt{  3})  }{  \cancel{2}(\sqrt{  2} + \sqrt{  3})  }

\longrightarrow \sf  \dfrac{ 2  \: \cancel{(\sqrt{    2} +  \sqrt{  3}) } }{  \cancel{(\sqrt{  2} + \sqrt{  3})}  }

\longrightarrow \sf   \green{2}

\star\:{\underline{\underline{\sf{\blue{Solution(2):}}}}}

\bullet  \sf \dfrac{ \sqrt{5}  +  \sqrt{3} }{ \sqrt{80} +  \sqrt{48}  - \sqrt{45}  -  \sqrt{27}   }

\begin{gathered}  \longrightarrow \sf  \dfrac{ \sqrt{5}  +  \sqrt{3} }{ \sqrt{4 \times 4 \times 5} +  \sqrt{4 \times 4 \times 3}  - \sqrt{3 \times 3 \times 5}  -  \sqrt{3 \times 3 \times 3}   } \end{gathered}

\begin{gathered}\longrightarrow  \sf \dfrac{ \sqrt{5}  +  \sqrt{3} }{ \sqrt{ {4}^{2}  \times 5} +  \sqrt{ {4}^{2}  \times 3}  - \sqrt{ {3}^{2}  \times 5}  -  \sqrt{ {3}^{2}  \times 3}   } \end{gathered}

\longrightarrow \sf  \dfrac{ \sqrt{5}  +  \sqrt{3} }{4 \sqrt{  5} + 4 \sqrt{   3}  - 3\sqrt{    5}  -  3\sqrt{  3}   }

\longrightarrow \sf  \dfrac{ \sqrt{5}  +  \sqrt{3} }{4 \sqrt{  5}   - 3\sqrt{    5} + 4 \sqrt{   3} -  3\sqrt{  3}   }

\longrightarrow  \sf \dfrac{ \cancel{ \sqrt{5}  +  \sqrt{3}} }{ \cancel{\sqrt{    5}  +   \sqrt{  3}   } }

\longrightarrow   \blue{1}

{\large{\textsf{\textbf{\underline{\underline{Answers :}}}}}}

• Question 1 - \purple{1}

• Question 2 - \red{250}

• Question 3(1) - \green{2}

• Question 3(2) - \blue{1}

{\large{\textsf{\textbf{\underline{\underline{ Concept \: :}}}}}}

<u>★</u><u> </u><u>Negative</u><u> Exponents Law -</u>

\bullet  \sf \:  {a}^{ - m}  =  \dfrac{1}{ {a}^{m} }

★ \sqrt{32} can be written as 4 \sqrt{2}

‣ \sqrt{48} can be written as 4 \sqrt{3}

‣ \sqrt{8} can be written as 2 \sqrt{2}

‣ \sqrt{12} can be written as 2 \sqrt{3}

‣ \sqrt{80} can be written as 4 \sqrt{5}

‣ \sqrt{48} can be written as 4 \sqrt{3}

‣ \sqrt{45} can be written as 3 \sqrt{5}

‣ \sqrt{27} can be written as 3 \sqrt{3}

★ <u>During Addition and Subtraction</u>

• minus (-) minus (-) gives plus (+)

• minus (-) plus (+) gives minus (-)

• plus (+) minus (-) gives minus (-)

• plus (+) plus (+) gives plus (+)

• Also the sign of the resultant term depends upon the sign of the largest number.

{\large{\textsf{\textbf{\underline{\underline{ Note \: :}}}}}}

• Swipe to see the full answer.

\begin{gathered} {\underline{\rule{330pt}{3pt}}} \end{gathered}

5 0
2 years ago
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