8A F11 Q10 - Revision history

Matt Lee at 22:36, 6 April 2015

2015-04-06T22:36:25Z

Matt Lee at 22:36, 6 April 2015

2015-04-06T22:36:05Z

Matt Lee at 22:35, 6 April 2015

2015-04-06T22:35:23Z

Matt Lee: Created page with "'''Question: ''' Graph the function. Give equations of any asymptotes, and list any intercepts $y = \frac{x-1}{2x+2}$ {| class="mw-collapsible mw-collapsed" style..."

2015-03-25T03:44:20Z

Created page with "'''Question: ''' Graph the function. Give equations of any asymptotes, and list any intercepts <math>y = \frac{x-1}{2x+2}</math> {| class="mw-collapsible mw-collapsed" style..."

New page

'''Question: ''' Graph the function. Give equations of any asymptotes, and list any intercepts <math>y = \frac{x-1}{2x+2}</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations
|-
|1) What are the asymptotes and zeros?
|-
|Answer:
|-
|1) The vertical asymptote corresponds to zeros of the denominator. So there is a vertical asymptote at x = -1. The zero is at (1, 0). The horizontal asymptote is the ratio of leading coefficients. So the horizontal asymptote is <math>y = \frac{1}{2}</math>
|}

Solution:

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 1:
|-
|We start by finding the asymptotes. The vertical asymptote corresponds to zeros of the denominator. So the vertical asymptote is at x = -1. They horizontal asymptote is determined by degree of the numerator and degree of the denominator. Since both of those values are 1, the horizontal asymptote is the ratio of leading coefficients. This means the horizontal asymptote is <math>y = \frac{1}{2}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 2:
|-
|Now we observe that the zero is at (1, 0), and proceed by looking at the intervals created by removing x = -1 and x = 1. This creates 3 intervals: <math>(-\infty, -1)</math>, <math>(-1, 1)</math>, and <math>(1, \infty)</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 3:
|-
|Now pick a number from each interval: -2, 0, 2 and find the value of the function for each number selected.
|-
|<math>x = -2:</math>     <math>\frac{-2 -1}{2(-2) + 2} = \frac{3}{2}</math>
|-
|<math>x = 0:</math>     <math>\frac{-1}{2}</math>
|-
|<math>x = 2:</math>     <math>\frac{2 - 1}{2(2) + 2} = \frac{1}{6}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 4:
|-
|The last check is whether or not the function intersects its horizontal asymptote. So check: <math>\frac{1}{2}=\frac{x - 1}{2x + 2}</math>.
|- style = "text-align: center"
|<math>\frac{1}{2} = \frac{x - 1}{2x + 2}</math>
|- style = "text-align: center"
|<math>2x + 2 = 2(x - 1)</math>
|- style = "text-align:center"
|<math>2x + 2 = 2x - 2</math>
|-style = "text-align:center"
|<math>4 = 0</math>
|-
|Since this is absurd, the function never intersects its horizontal asymptote. Now we graph while respecting the asymptotes
|}

@@ Line 44: / Line 44: @@
+|
 <math>\begin{array}{rcl}
-\displaystyle{\frac{1}{2} &=& \frac{x - 1}{2x + 2}\\
+\frac{1}{2} &=& \frac{x - 1}{2x + 2}\\
 x + 2 &=& 2(x - 1)\\
 x + 2 &=& 2x - 2\\
-&=& 0}
+&=& 0
 \end{array}</math>
 |-

@@ Line 44: / Line 44: @@
+|
 <math>\begin{array}{rcl}
-\frac{1}{2} &=& \frac{x - 1}{2x + 2}\\
+\displaystyle{\frac{1}{2} &=& \frac{x - 1}{2x + 2}\\
 x + 2 &=& 2(x - 1)\\
 x + 2 &=& 2x - 2\\
-&=& 0
+&=& 0}
 \end{array}</math>
 |-

@@ Line 42: / Line 42: @@
 |The last check is whether or not the function intersects its horizontal asymptote. So check: <math>\frac{1}{2}=\frac{x - 1}{2x + 2}</math>.
 |- style = "text-align: center"
-|<math>\frac{1}{2} = \frac{x - 1}{2x + 2}</math>
-|- style = "text-align: center"
+<math>\begin{array}{rcl}
-|<math>2x + 2 = 2(x - 1)</math>
+\frac{1}{2} &=& \frac{x - 1}{2x + 2}\\
-|- style = "text-align:center"
+x + 2 &=& 2(x - 1)\\
-|<math>2x + 2 = 2x - 2</math>
+x + 2 &=& 2x - 2\\
-|-style = "text-align:center"
+&=& 0
-|<math>4 = 0</math>
+\end{array}</math>
 |-
 |Since this is absurd, the function never intersects its horizontal asymptote. Now we graph while respecting the asymptotes
 |}

8A F11 Q10 - Revision history

Matt Lee at 22:36, 6 April 2015

Matt Lee at 22:36, 6 April 2015

Matt Lee at 22:35, 6 April 2015

Matt Lee: Created page with "'''Question: ''' Graph the function. Give equations of any asymptotes, and list any intercepts y = \frac{x-1}{2x+2} {| class="mw-collapsible mw-collapsed" style..."

Matt Lee: Created page with "'''Question: ''' Graph the function. Give equations of any asymptotes, and list any intercepts $y = \frac{x-1}{2x+2}$ {| class="mw-collapsible mw-collapsed" style..."