005 Sample Final A, Question 19 - Revision history

MathAdmin at 01:22, 3 June 2015

2015-06-03T01:22:34Z

MathAdmin at 01:21, 3 June 2015

2015-06-03T01:21:33Z

MathAdmin at 19:50, 2 June 2015

2015-06-02T19:50:14Z

MathAdmin at 19:43, 2 June 2015

2015-06-02T19:43:36Z

MathAdmin at 19:05, 2 June 2015

2015-06-02T19:05:30Z

MathAdmin: Created page with "'''Question ''' Consider the following function, $f(x) = -\sin\left(3x+\frac{\pi}{2}\right)+1$
a. What is the amplitu..."

2015-06-01T01:37:26Z

Created page with "'''Question ''' Consider the following function, <center><math>f(x) = -\sin\left(3x+\frac{\pi}{2}\right)+1</math></center> a. What is the amplitu..."

New page

'''Question ''' Consider the following function, <center><math>f(x) = -\sin\left(3x+\frac{\pi}{2}\right)+1</math></center> 

     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.

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Final Answers
|-
|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>
|-
|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.
|-
|c) False. <math>y = x^2</math> does not have an inverse.
|-
|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
|-
|e) True.
|-
|f) False.
|}

[[005 Sample Final A|'''Return to Sample Exam''']]

@@ Line 47: / Line 47: @@
 ! Final Answer: &nbsp;
 |-
-|The final solution is the portion of the graph that below <math>y = \vert x\vert + 1</math> and inside <math> x^2 + y^2 = 9</math>
+|Amplitude: -1, period: <math>\frac{2\pi}{3}~</math>, phase shift: Left by <math>\frac{\pi}{2}~</math> and vertical shift up by 1.
 |-
-|The region we are referring to is shaded both blue and red.
+|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>
 |-
 |[[File:5_Sample_Final_19.png]]

@@ Line 35: / Line 35: @@
 ! Step 2: &nbsp;
 |-
-|The five key points for the graph are <math>(-\frac{\pi}{6}, 0)</math>
+|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>
 |}
@@ Line 45: / Line 51: @@
 |The region we are referring to is shaded both blue and red.
 |-
-|[[File:8A_Final_5.png]]
+|[[File:5_Sample_Final_19.png]]
 |}
 [[005 Sample Final A|'''<u>Return to Sample Exam</u>''']]

@@ Line 19: / Line 19: @@
 |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.
 |-
-|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>
+|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>
 |}
@@ Line 31: / Line 31: @@
 |Amplitude: -1, period: <math>\frac{2\pi}{3}~</math>, phase shift: Left by <math>\frac{\pi}{2}~</math> and vertical shift up by 1.
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 ! Step 2: &nbsp;
 |-
-|Now that we have graphed both functions we need to know which region to shade with respect to each graph.
+|The five key points for the graph are <math>(-\frac{\pi}{6}, 0)</math>
 |}

@@ Line 19: / Line 19: @@
 |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.
 |-
-|2) Since the Y-value must be less than <math>\vert x\vert + 1</math>, shade below the V. For the circle shde the inside.
+|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>
 |}
@@ Line 27: / Line 27: @@
 ! Step 1: &nbsp;
 |-
-|First we replace the inequalities with equality. So <math>y = \vert x\vert + 1</math>, and <math>x^2 + y^2 = 9</math>.
+|We can read off the answers for (a) - (d):
 |-
-|Now we graph both functions.
+|Amplitude: -1, period: <math>\frac{2\pi}{3}~</math>, phase shift: Left by <math>\frac{\pi}{2}~</math> and vertical shift up by 1.
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"

@@ Line 1: / Line 1: @@
 '''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>
+:: a. What is the amplitude?<br>
-&nbsp;&nbsp;&nbsp;&nbsp; b. What is the period? <br>
+:: b. What is the period? <br>
-&nbsp;&nbsp;&nbsp;&nbsp; c. What is the phase shift? <br>
+:: c. What is the phase shift? <br>
-&nbsp;&nbsp;&nbsp;&nbsp; d. What is the vertical shift? <br>
+:: 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.
+:: e. Graph one cycle of f(x). Make sure to label five key points.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-! Final Answers
+! 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>
+|1) For parts (a) - (d), How do we read the relevant information off of <math>A\sin(Bx + C) + D?</math>
 |-
-|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.
+|2) What are the five key points when looking at <math>\sin(x)?</math>
 |-
-|c) False. <math>y = x^2</math> does not have an inverse.
+|Answer:
 |-
-|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
+|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.
 |-
-|e) True.
+|2) Since the Y-value must be less than <math>\vert x\vert + 1</math>, shade below the V. For the circle shde the inside.
 |-
-|f) False.
+|We also shade the boundary of the circle since the inequality is <math>\le</math>
 |}
 [[005 Sample Final A|'''<u>Return to Sample Exam</u>''']]