From 240912d7ad79bebb20724c506b0040ee1dbb1042 Mon Sep 17 00:00:00 2001
From: kathypinzon <75687638+kathypinzon@users.noreply.github.com>
Date: Sun, 28 Jul 2024 22:58:22 +0000
Subject: [PATCH] almost all answers to PR6

---
 source/04-PR/06.ptx | 244 ++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 236 insertions(+), 8 deletions(-)

diff --git a/source/04-PR/06.ptx b/source/04-PR/06.ptx
index be4d5817..36cdc01c 100644
--- a/source/04-PR/06.ptx
+++ b/source/04-PR/06.ptx
@@ -40,12 +40,24 @@
         <statement>
         <p>Find <m>r(1) </m>, <m>r(2) </m>, <m>r(3)</m>, and <m>r(4)</m>.  </p>
       </statement>
+      <answer>
+        <p>
+          <m>r(1)</m> and <m>r(3)</m> do not exist.
+          <m>r(2)=0</m>
+          <m>r(4)=2</m>
+        </p>
+      </answer>
       </task>
       <task>
         <statement>
-      <p> Label each of these four points as giving us information about the DOMAIN of <m>r(x)</m>, information about the ZEROES of <m>r(x)</m>, or NEITHER.    
+      <p> Label each of these four values as giving us information about the DOMAIN of <m>r(x)</m>, information about the ZEROES of <m>r(x)</m>, or NEITHER.    
       </p>
     </statement>
+    <answer>
+      <p>
+        <m>r(1)</m> and <m>r(3)</m> give information about the domain.  <m>r(2)</m> gives information about the zeroes. <m>r(4)</m> does not give information about either.
+      </p>
+    </answer>
     </task>
       
     </activity>
@@ -74,6 +86,11 @@
         <statement>
         <p>Rewrite <m>r(x)</m> by factoring the numerator and denominator, but do not try to simplify any further. What do you notice about the relationship between the values that are not in the domain and how the function is now written? </p>
       </statement>
+      <answer>
+        <p>
+          <m>r(x)=\dfrac{(x-2)(x-1)}{(x-3)(x-1)}</m> The values not in the domain are zeros of the denominator.
+        </p>
+      </answer>
       </task>
       
       <task>
@@ -113,6 +130,50 @@
             </tabular>
           </table> </p>
         </statement>
+        <answer>
+          <p>
+            
+              <tabular>
+                <row header="yes" bottom="minor" >
+                  <cell><m>x</m></cell> 
+                  <cell><m>r(x)</m></cell>
+                </row>
+                <row halign="center">
+                  <cell><m>2</m></cell>
+                  <cell><m>0</m></cell>
+                </row>
+                <row halign="center">
+                  <cell><m>2.9</m></cell>
+                  <cell><m>-9</m></cell>
+                </row>
+                <row halign="center">
+                  <cell><m>2.99</m></cell>
+                  <cell><m>-99</m></cell>
+                </row>
+                <row halign="center">
+                  <cell><m>2.999</m></cell>
+                  <cell><m>-999</m></cell>
+                </row>
+                <row halign="center">
+                  <cell><m>3</m></cell>
+                  <cell>undefined</cell>
+                </row>
+                <row halign="center">
+                  <cell><m>3.001</m></cell>
+                  <cell><m>1001</m></cell>
+                </row>
+                <row halign="center">
+                  <cell><m>3.01</m></cell>
+                  <cell><m>101</m></cell>
+                </row>
+                <row halign="center">
+                  <cell><m>3.1</m></cell>
+                  <cell><m>11</m></cell>
+                </row>
+              </tabular>
+            
+          </p>
+        </answer>
       </task>
       <task>
         <statement>
@@ -125,6 +186,11 @@
         <li> As <m>x \to 3</m> from the right, <m>r(x)\to -\infty</m></li></ol>
       </p>
     </statement>
+    <answer>
+      <p>
+        C and D
+      </p>
+    </answer>
       </task>
       <task>
         <statement>
@@ -163,6 +229,50 @@
             </tabular>
           </table> </p>
       </statement>
+      <answer>
+        <p>
+          
+            <tabular>
+              <row header="yes" bottom="minor" >
+                <cell><m>x</m></cell> 
+                <cell><m>r(x)</m></cell>
+              </row>
+              <row halign="center">
+                <cell><m>0</m></cell>
+                <cell><m>0.667</m></cell>
+              </row>
+              <row halign="center">
+                <cell><m>0.9</m></cell>
+                <cell><m>0.524</m></cell>
+              </row>
+              <row halign="center">
+                <cell><m>0.99</m></cell>
+                <cell><m>0.502</m></cell>
+              </row>
+              <row halign="center">
+                <cell><m>0.999</m></cell>
+                <cell><m>0.500</m></cell>
+              </row>
+              <row halign="center">
+                <cell><m>1</m></cell>
+                <cell>undefined</cell>
+              </row>
+              <row halign="center">
+                <cell><m>1.001</m></cell>
+                <cell><m>0.499</m></cell>
+              </row>
+              <row halign="center">
+                <cell><m>1.01</m></cell>
+                <cell><m>0.497</m></cell>
+              </row>
+              <row halign="center">
+                <cell><m>1.1</m></cell>
+                <cell><m>0.474</m></cell>
+              </row>
+            </tabular>
+          
+        </p>
+      </answer>
       </task>
       <task>
         <statement>
@@ -175,11 +285,21 @@
         <li> As <m>x \to 1</m> from the right, <m>r(x)\to -\infty</m></li></ol>
       </p>
     </statement>
+    <answer>
+      <p>
+        A
+      </p>
+    </answer>
       </task>
       <task>
         <statement>
         <p>The function is behaving differently near <m>x=1</m> than it is near <m>x=3</m>. Can you see anything in the factored form of <m>r(x)</m> that may help you account for the difference? </p>
       </statement>
+      <answer>
+        <p>
+          The <m>x-1</m> factor occurs in the numerator and denominator.
+        </p>
+      </answer>
       </task>
     </activity>
 <!-- 
@@ -279,6 +399,11 @@
             </ol> 
           </p>
         </statement>
+        <answer>
+          <p>
+            C
+          </p>
+        </answer>
         </task>
 
 
@@ -294,6 +419,11 @@
             </ol> 
           </p>
         </statement>
+        <answer>
+          <p>
+            C
+          </p>
+        </answer>
         </task>
       
 
@@ -352,6 +482,11 @@
             </ol> 
           </p>
         </statement>
+        <answer>
+          <p>
+            B
+          </p>
+        </answer>
         </task>
 
         <task>
@@ -365,6 +500,11 @@
                   Use <xref ref="obs-rf-HA"></xref> to help!
                 </p>
               </hint>
+              <answer>
+                <p>
+                  Numerator goes to <m>3</m> and denominator goes to <m>7</m>.
+                </p>
+              </answer>
         </task>
 
         <task>
@@ -385,6 +525,11 @@
 
             </p>
           </statement>
+          <answer>
+            <p>
+              B
+            </p>
+          </answer>
         </task>
       </activity>
 
@@ -423,6 +568,11 @@
           </ol>  
           </p>
       </statement>
+      <answer>
+        <p>
+          B
+        </p>
+      </answer>
       </task>
       <task>       
         <statement>
@@ -442,6 +592,11 @@
           </li>       
         </ol>
     </statement>
+    <answer>
+      <p>
+        A
+      </p>
+    </answer>
     </task>
       <task>       
         <statement>
@@ -461,6 +616,11 @@
         </ol>
         </p>
     </statement>
+    <answer>
+      <p>
+        E
+      </p>
+    </answer>
     </task>
     
     </activity> 
@@ -478,6 +638,11 @@
            If the degree of the numerator is the same as the degree of the denominator, then... 
           </p>
       </statement>
+      <answer>
+        <p>
+          the horizontal asymptote is a non-zero number.
+        </p>
+      </answer>
       </task>
       <task>
         <statement>
@@ -485,6 +650,11 @@
          If the degree of the numerator is less than the degree of the denominator, then... 
         </p>
       </statement>
+      <answer>
+        <p>
+          there is a horizontal asymptote at <m>0</m>.
+        </p>
+      </answer>
       </task>
       <task>
         <statement>
@@ -492,6 +662,11 @@
          If the degree of the numerator is greater than the degree of the denominator, then... 
         </p>
       </statement>
+      <answer>
+        <p>
+          there is no horizontal asymptote.
+        </p>
+      </answer>
       </task>
     </activity>  
 
@@ -575,11 +750,21 @@
         <statement>
         <p>Which of the graphs above represents the function <m>\displaystyle f(x)=\frac{2x}{x^2-2x-8}</m>?</p>
       </statement>
+      <answer>
+        <p>
+          Graph A
+        </p>
+      </answer>
       </task>
       <task>
         <statement>
         <p>Which of the graphs above represents the function <m>\displaystyle g(x)=\frac{x^2+3}{2x^2-8}</m>?</p>
         </statement>
+        <answer>
+          <p>
+            Graph E
+          </p>
+        </answer>
       </task>
     </activity>
 
@@ -675,11 +860,21 @@
         <statement>
         <p>Which of the graphs above represents the function <m>\displaystyle f(x)=\frac{x^2+5x+6}{(x^2-3x-10)(x+4)}</m>?</p>
         </statement>
+        <answer>
+          <p>
+            Graph A
+          </p>
+        </answer>
       </task>
       <task>
         <statement>
         <p>Which of the graphs above represents the function <m>\displaystyle g(x)=\frac{x^2-3x-10}{(x+2)(x^2-x-20)}</m>?</p>
       </statement>
+      <answer>
+        <p>
+          Graph E
+        </p>
+      </answer>
       </task>
     </activity> 
  
@@ -693,6 +888,11 @@
         <statement>
         <p> Find the roots of <m>f(x)</m>. </p>
       </statement>
+      <answer>
+        <p>
+          <m>4</m>
+        </p>
+      </answer>
       </task>
 
       <task>
@@ -700,11 +900,21 @@
         <p> Find the <m>y</m>-intercept of the graph of <m>f(x)</m>. </p>
   
         </statement>
+        <answer>
+          <p>
+            <m>\left(0,\dfrac{2}{9}\right)</m>
+          </p>
+        </answer>
       </task>
       <task>
         <statement>
           <p> Find any horizontal asymptotes on the graph of  <m>f(x)</m>. </p>
         </statement>
+        <answer>
+          <p>
+            <m>y=0</m>
+          </p>
+        </answer>
        
       </task>
 
@@ -712,25 +922,33 @@
         <statement>
           <p> Find any vertical asymptotes on the graph of <m>f(x)</m>. </p>
         </statement>
-        <!-- <answer>
+        <answer>
           <p> <m>x=-3</m>. </p>
-        </answer> -->
+        </answer>
       </task>
       <task>
         <statement>
           <p> Find any holes on the graph of <m>f(x)</m>. </p>
         </statement>
-        <!-- <answer>
+        <answer>
           <p> There is a hole at <m>\left(1, \frac{3}{32} \right)</m>. </p>
-        </answer> -->
+        </answer>
       </task>
       <task>
         <statement>
           <p> Sketch the graph of <m>f(x)</m>. </p>
         </statement>
-        <!-- <answer>
-          <p> <image source="assets/PR6/generated/0019/plot.png" description="The plot of y=f(x), along with special points."/> </p>
-        </answer> -->
+        <answer>
+          <p> <image>
+            <sageplot>
+              f(x) = (-(x-1)*(x-4))/(2*(x-1)*(x+3)^2)
+              p=plot(f, (x, -7, 7), ymin=-7, ymax=7, color='blue', thickness=3, detect_poles=True)
+              p+=parametric_plot((-3,x),(x,-8,8),color='red',linestyle="--")
+              p+=parametric_plot((x,0),(x,-7,7),color='red',linestyle="--")
+              p 
+            </sageplot>
+          </image>  </p>
+        </answer> 
       </task>
    
     </activity> 
@@ -749,6 +967,11 @@ https://activecalculus.org/prelude/sec-poly-rational-features.html#ez-poly-ratio
           <m>\displaystyle r(x) = \frac{x^2-16}{x+4}</m>
         </p>
       </statement>
+      <answer>
+        <p>
+          At <m>x=-4</m> there is a hole since the function values tend towards a number.
+        </p>
+      </answer>
       </task>
       <task>
         <statement>
@@ -756,6 +979,11 @@ https://activecalculus.org/prelude/sec-poly-rational-features.html#ez-poly-ratio
           <m>\displaystyle s(x) = \frac{(x-2)^2(x+3)}{x^2 - 5x - 6}</m>
         </p>
       </statement>
+      <answer>
+        <p>
+          There are no holes.
+        </p>
+      </answer>
       </task>
       <task>
         <statement>