022 Exam 2 Sample B, Problem 2 - Revision history

MathAdmin at 16:04, 18 May 2015

2015-05-18T16:04:51Z

MathAdmin at 00:16, 18 May 2015

2015-05-18T00:16:51Z

MathAdmin at 00:06, 18 May 2015

2015-05-18T00:06:12Z

MathAdmin at 23:37, 17 May 2015

2015-05-17T23:37:10Z

MathAdmin at 20:11, 17 May 2015

2015-05-17T20:11:25Z

MathAdmin: Created page with " Sketch the graph of $y = \left(\frac{1}{2}\right)^{x + 1} - 4$ . {| class="mw-collapsible mw-collapsed" style = "te..."

2015-05-17T20:10:48Z

Created page with "<span class="exam"> Sketch the graph of <math style="vertical-align: -52%">y = \left(\frac{1}{2}\right)^{x + 1} - 4</math>. {| class="mw-collapsible mw-collapsed" style = "te..."

New page

<span class="exam"> Sketch the graph of <math style="vertical-align: -52%">y = \left(\frac{1}{2}\right)^{x + 1} - 4</math>.

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|This is a problem about graphing through transformations. It requires you to find the basic or prototype graph, and then understand how to apple the transformations. In particular, if our basic graph is
|-
|
::<math>y\,=\,f(x),</math>
|-
|and we have a transformed graph
|-
|
::<math>y=A\,f(x-B)+C,</math>
|-
|we would have to consider a shift/mirroring of the basic graph from <math style="vertical-align: 0%">A</math>, a horizontal shift from <math style="vertical-align: 0%">B</math>, and a vertical shift from <math style="vertical-align: 0%">C</math>.
|}

 '''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1: 
|-
|'''Identify the Basic Graph:''' The basic graph is
|-
|
::<math>y=\left(\frac{1}{2}\right)^x.</math>
|-
|If you do not know exactly what this looks like, plot the basic points:
|-
|
::<math>\begin{array}{|c||c|c|c|c|c|}
\hline x & -2\, & -1\, & \,\,0\,\, & 1 & 2\\
\hline (1/2)^{x} & \,4 & \,2 & 1 & 1/2 & 1/4
\\\hline \end{array}</math>
|-
|I would always recommend plotting the basic graph, in order to show that you properly applied the transformations. Note that since our base is less than one, the basic graph will be decreasing.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2: 
|-
|'''Verify the Transformations:''' Here, we need to shift the basic graph down by four, while moving it to the right one (as the argument <math style="vertical-align: -10%">x+1</math> is zero when <math style="vertical-align: 0%">x</math> is one). Note that since the basic graph has an asymptote at the <math style="vertical-align: 0%">x</math>-axis, we will shift the asymptote to the line <math style="vertical-align: -18%">y=-4.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|
::<math>x^3-6x^2+8x+C.</math>
|}
[[022_Exam_2_Sample_B|'''<u>Return to Sample Exam</u>''']]

@@ Line 47: / Line 47: @@
 !Final Answer: &nbsp;
 |-
-|In addition to your final graph, for grading purposes you should show your basic graph, the new asymptote and the translations of a few points. The red dots show the values for the basic graph from the chart in step 1.
+|In addition to your final graph, for grading purposes you should show your basic graph, the new asymptote and the translations of a few points. The red dots show the values for the basic graph from the chart in step 1.  Many teachers will also ask that you label a few points on your final graph (shown here in blue).
 <br>
 |-

@@ Line 14: / Line 14: @@
 ::<math>y=A\,f(x-B)+C,</math>
 |-
-|we would have to consider a shift/mirroring of the basic graph from <math style="vertical-align: 0%">A</math>, a horizontal shift from <math style="vertical-align: 0%">B</math>, and a vertical shift from <math style="vertical-align: 0%">C</math>.
+|we would have to consider a vertical scaling/mirroring of the basic graph from <math style="vertical-align: 0%">A</math>, a horizontal shift from <math style="vertical-align: 0%">B</math>, and a vertical shift from <math style="vertical-align: 0%">C</math>.
 |}

@@ Line 41: / Line 41: @@
 !Step 2:&nbsp;
 |-
-|'''Verify the Transformations:''' Here, we need to shift the basic graph down by four, while moving it to the left one (as the argument <math style="vertical-align: -8%">x+1</math> is zero when <math style="vertical-align: 0%">x=-1</math>).  Note that since the basic graph has an asymptote at the <math style="vertical-align: 0%">x</math>-axis, we will shift the asymptote to the line <math style="vertical-align: -16%">y=-4.</math>
+|'''Verify the Transformations:''' Here, we need to shift the basic graph down by four, while moving it to the left one (as the argument <math style="vertical-align: -8%">x+1</math> is zero when <math style="vertical-align: -5%">x=-1</math>).  Note that since the basic graph has an asymptote at the <math style="vertical-align: 0%">x</math>-axis, we will shift the asymptote to the line <math style="vertical-align: -16%">y=-4.</math>
 |}
@@ Line 47: / Line 47: @@
 !Final Answer: &nbsp;
 |-
-|Not done yet!
+|In addition to your final graph, for grading purposes you should show your basic graph, the new asymptote and the translations of a few points. The red dots show the values for the basic graph from the chart in step 1.
 |}
 [[022_Exam_2_Sample_B|'''<u>Return to Sample Exam</u>''']]

@@ Line 4: / Line 4: @@
 !Foundations: &nbsp;
 |-
-|This is a problem about graphing through transformations.  It requires you to find the basic or prototype graph, and then understand how to apple the transformations. In particular, if our basic graph is
+|This is a problem about graphing through transformations.  It requires you to find the basic or prototype graph, and then understand how to apply the transformations. In particular, if our basic graph is
 |-
+|
@@ Line 41: / Line 41: @@
 !Step 2:&nbsp;
 |-
-|'''Verify the Transformations:''' Here, we need to shift the basic graph down by four, while moving it to the right one (as the argument <math style="vertical-align: -8%">x+1</math> is zero when <math style="vertical-align: 0%">x</math> is one).  Note that since the basic graph has an asymptote at the <math style="vertical-align: 0%">x</math>-axis, we will shift the asymptote to the line <math style="vertical-align: -16%">y=-4.</math>
+|'''Verify the Transformations:''' Here, we need to shift the basic graph down by four, while moving it to the left one (as the argument <math style="vertical-align: -8%">x+1</math> is zero when <math style="vertical-align: 0%">x=-1</math>).  Note that since the basic graph has an asymptote at the <math style="vertical-align: 0%">x</math>-axis, we will shift the asymptote to the line <math style="vertical-align: -16%">y=-4.</math>
 |}