Difference between revisions of "005 Sample Final A, Question 19"

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(Created page with "'''Question ''' Consider the following function, <center><math>f(x) = -\sin\left(3x+\frac{\pi}{2}\right)+1</math></center><br>      a. What is the amplitu...")
 
 
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'''Question ''' Consider the following function, <center><math>f(x) = -\sin\left(3x+\frac{\pi}{2}\right)+1</math></center><br>
 
'''Question ''' Consider the following function, <center><math>f(x) = -\sin\left(3x+\frac{\pi}{2}\right)+1</math></center><br>
  
&nbsp;&nbsp;&nbsp;&nbsp; a. What is the amplitude?<br>
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:: a. What is the amplitude?<br>
&nbsp;&nbsp;&nbsp;&nbsp; b. What is the period? <br>
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:: b. What is the period? <br>
&nbsp;&nbsp;&nbsp;&nbsp; c. What is the phase shift? <br>
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:: c. What is the phase shift? <br>
&nbsp;&nbsp;&nbsp;&nbsp; d. What is the vertical shift? <br>
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:: d. What is the vertical shift? <br>
&nbsp;&nbsp;&nbsp;&nbsp; e. Graph one cycle of f(x). Make sure to label five key points.
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:: e. Graph one cycle of f(x). Make sure to label five key points.
  
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Final Answers
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! Foundations: &nbsp;
 
|-
 
|-
|a) False. Nothing in the definition of a geometric sequence requires the common ratio to be always positive. For example, <math>a_n = (-a)^n</math>
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|1) For parts (a) - (d), How do we read the relevant information off of <math>A\sin(Bx + C) + D?</math>
 
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|b) False. Linear systems only have a solution if the lines intersect. So y = x and y = x + 1 will never intersect because they are parallel.
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|2) What are the five key points when looking at <math>\sin(x)?</math>
 
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|c) False. <math>y = x^2</math> does not have an inverse.
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|Answer:
 
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|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
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|1) The amplitude is A, the period is <math>\frac{2\pi}{B}</math>, the horizontal shift is left by C units if C is positive and right by C units if C is negative, the vertical shift is up by D if D is positive and down by D units if D is negative.  
 
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|e) True.
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|2) The five key points are <math>(0, 0),~ (\frac{\pi}{2}, 1), ~ (\pi, 0), ~ (\frac{3\pi}{2}, 0),~ \text{and } (2\pi, 0).</math>
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|}
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Solution:
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 1: &nbsp;
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|-
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|We can read off the answers for (a) - (d):
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|-
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|Amplitude: -1, period: <math>\frac{2\pi}{3}~</math>, phase shift: Left by <math>\frac{\pi}{2}~</math> and vertical shift up by 1.
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 2: &nbsp;
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|-
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|The five key points for the graph are <math>(-\frac{\pi}{6}, 1), (0, 0), (\frac{\pi}{6}, 1), (\frac{\pi}{3}, 2), (\frac{\pi}{2}, 1)</math>
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 3: &nbsp;
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|-
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|Now plot the five key points and sketch a graph that looks like the <math>\sin</math> graph through those points.
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Final Answer: &nbsp;
 
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|f) False.  
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|Amplitude: -1, period: <math>\frac{2\pi}{3}~</math>, phase shift: Left by <math>\frac{\pi}{2}~</math> and vertical shift up by 1.
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|-
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|The five key points for the graph are <math>(-\frac{\pi}{6}, 1), (0, 0), (\frac{\pi}{6}, 1), (\frac{\pi}{3}, 2), (\frac{\pi}{2}, 1)</math>
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|-
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|[[File:5_Sample_Final_19.png]]
 
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|}
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[[005 Sample Final A|'''<u>Return to Sample Exam</u>''']]
 
[[005 Sample Final A|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 18:22, 2 June 2015

Question Consider the following function,


a. What is the amplitude?
b. What is the period?
c. What is the phase shift?
d. What is the vertical shift?
e. Graph one cycle of f(x). Make sure to label five key points.


Foundations:  
1) For parts (a) - (d), How do we read the relevant information off of
2) What are the five key points when looking at
Answer:
1) The amplitude is A, the period is , the horizontal shift is left by C units if C is positive and right by C units if C is negative, the vertical shift is up by D if D is positive and down by D units if D is negative.
2) The five key points are

Solution:

Step 1:  
We can read off the answers for (a) - (d):
Amplitude: -1, period: , phase shift: Left by and vertical shift up by 1.
Step 2:  
The five key points for the graph are
Step 3:  
Now plot the five key points and sketch a graph that looks like the graph through those points.
Final Answer:  
Amplitude: -1, period: , phase shift: Left by and vertical shift up by 1.
The five key points for the graph are
5 Sample Final 19.png


Return to Sample Exam