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Second attempt to fix FN3 Tweaks #180 (#229)
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@tdegeorge
tdegeorge authored May 20, 2024
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105 changes: 86 additions & 19 deletions source/02-FN/03.ptx
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<sageplot>
f(x) = -1/2*x-2
p=plot(f, (x, -5, 5), ymin=-10, ymax=10, color='blue', thickness=3)
p+=text('Graph A', (2, 8), fontsize=18, color='black')
p+=text('f(x)', (4,-2), fontsize=18, color='black')
p
</sageplot>
</image>
<image>
<sageplot>
f(x) = (x+1)*(x-3)
p=plot(f, (x, -5, 5), ymin=-10, ymax=10, color='blue', thickness=3)
p+=text('Graph B', (2, 8), fontsize=18, color='black')
p+=text('g(x)', (4, 1), fontsize=18, color='black')
p
</sageplot>
</image>
</sidebyside>
</introduction>
<task>
<statement>
<p> What are the <m>x</m>-intercept(s) of Graph A?
<p> What are the <m>x</m>-intercept(s) of <m>f(x)</m>?
<ol marker= "A." cols="2">
<li> <m>(0, -4)</m> </li>
<li> <m>(-2, 0)</m> </li>
Expand All @@ -65,7 +65,7 @@
</task>
<task>
<statement>
<p> What are the <m>x</m>-intercept(s) of Graph B?
<p> What are the <m>x</m>-intercept(s) of <m>g(x)</m>?
<ol marker= "A." cols="2">
<li> <m>(0, -3)</m> </li>
<li> <m>(-1, 0)</m> </li>
Expand All @@ -80,7 +80,7 @@
</task>
<task>
<statement>
<p> What are the <m>y</m>-intercept(s) of Graph A?
<p> What are the <m>y</m>-intercept(s) of <m>f(x)</m>?
<ol marker= "A." cols="2">
<li> <m>(0, -4)</m> </li>
<li> <m>(-2, 0)</m> </li>
Expand All @@ -95,7 +95,7 @@
</task>
<task>
<statement>
<p> What are the <m>y</m>-intercept(s) of Graph B?
<p> What are the <m>y</m>-intercept(s) of <m>g(x)</m>?
<ol marker= "A." cols="2">
<li> <m>(0, -3)</m> </li>
<li> <m>(-1, 0)</m> </li>
Expand All @@ -111,7 +111,7 @@
<task>
<statement>
<p>
Suppose you are told that a function has the following intercepts:
Sketch a graph of a function with the following intercepts:
<ul>
<li>
<m>x</m>-intercepts: <m>(-2,0)</m> and <m>(6,0)</m>
Expand All @@ -121,8 +121,6 @@
</li>
</ul>
</p>
<p>What would the graph look like?
</p>
</statement>
<answer>
<p>
Expand All @@ -133,12 +131,20 @@
<task>
<statement>
<p>
What would a graph look like if it had more than one <m>y</m>-intercept?
Sketch a graph of a function with the following intercepts:
<ul>
<li>
<m>x</m>-intercept: <m>(-1,0)</m>
</li>
<li>
<m>y</m>-intercept: <m>(0,6)</m> and <m>(0,-2)</m>
</li>
</ul>
</p>
</statement>
<answer>
<p>
You could draw a variety of graphs, but they all would not be functions.
You could draw a variety of graphs, but they would not be functions.
</p>
</answer>
</task>
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<sageplot>
f(x) = 9/8*(x-4)+4
p=plot(f, (x, -4, 4), ymin=-7, ymax=7, xmin=-5,xmax=5,color='blue', thickness=3,gridlines=[[-8..8],[-8..8]])
p+=point((4,4),pointsize=50,color='blue')
p+=point((-4,-5),pointsize=50,color='blue')
p+=point((4,4),pointsize=75,color='blue')
p+=point((-4,-5),pointsize=75,color='blue')
p
</sageplot>
</image>
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<image>
<sageplot>
p=arrow((0,4), (9,-2),xmin=-2,xmax=9,ymin=-5,ymax=6, gridlines=[[-8..8],[-8..8]])
p+=point((0,4),pointsize=50,color='blue')
p+=point((0,4),pointsize=75,color='blue')
p
</sageplot>
</image>
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<image>
<sageplot>
p=arrow((3,-4), (-4,8),xmin=-6,xmax=5,ymin=-6,ymax=8, gridlines=[[-8..8],[-8..8]])
p+=point((3,-4),pointsize=50,color='blue')
p+=point((3,-4),pointsize=10,color='white',zorder=2)
p+=point((3,-4), pointsize=75, markeredgecolor='blue',color='white',zorder=2)
p
</sageplot>
</image>
Expand Down Expand Up @@ -352,9 +357,8 @@
p=line([(5,-1), (4,-5)],thickness=3,xmin=-6,xmax=8,ymin=-6,ymax=8, gridlines=[[-8..8],[-8..8]])
p+=line([(4,-5),(-1,6)],thickness=3)
p+=line([(-1,6),(-3,1)],thickness=3)
p+=point((5,-1),pointsize=50,color='blue')
p+=point((5,-1),pointsize=10,color='white',zorder=2)
p+=point((-3,1),pointsize=50,color='blue')
p+=point((5,-1),pointsize=50,markeredgecolor='blue',color='white',zorder=2)
p
</sageplot>
</image>
Expand Down Expand Up @@ -492,7 +496,7 @@
The <term>maximum value</term>, or <term>global maximum</term>, of a graph is the point where the <m>y</m>-coordinate has the largest value. The <term>minimum value</term>, or <term>global minimum</term> is the point on the graph where the <m>y</m>-coordinate has the smallest value.
</p>
<p>
Graphs can also have <term>relative/local maximums</term> and <term>relative/local minimums</term>. A relative maximum point is a point where the function changes direction from increasing to decreasing (making that point a "hill" in the graph). Similarly, a relative minimum point is a point where the function changes direction from decreasing to increasing (making that point a "valley" in the graph).
Graphs can also have <term>relative/local maximums</term> and <term>relative/local minimums</term>. A relative maximum point is a point where the function value (i.e, <m>y</m>-value) is larger than all others in some neighborhood around the point. Similarly, a relative minimum point is a point where the function value (i.e, <m>y</m>-value) is smaller than all others in some neighborhood around the point.
</p>
</statement>
</definition>
Expand Down Expand Up @@ -649,6 +653,69 @@
</p>
</remark>

<activity xml:id="extension-activity-investigating-minima">
<introduction>
<p>
Sometimes, it is not always clear what the maxima or minima are or if they exist. Consider the following graph of <m>f(x)</m>:
</p>
<image>
<sageplot>
f(x) = abs(1*x)
p=plot(f, (x, -5, 5), ymin=-5, ymax=5, color='blue', thickness=3)
p+=point((0,0), pointsize=75, markeredgecolor='blue',color='white',zorder=3)
p+=point((0,1),pointsize=75,color='blue')
p
</sageplot>
</image>
</introduction>
<task>
<statement>
<p>
What is the value of <m>f(0)</m>?
<ol marker= "A." cols="1">
<li> <m>1</m> </li>
<li> <m>0</m> </li>
<li> There is no relative minimum </li></ol></p>
</statement>
<answer>
<p>
A
</p>
</answer>
</task>

<task>
<statement>
<p>
What is the relative minimum of <m>f(x)</m>?
<ol marker= "A." cols="1">
<li> <m>1</m> </li>
<li> <m>0</m> </li>
<li> There is no relative minimum </li></ol></p>
</statement>
<answer>
<p>
C
</p>
</answer>
</task>
<task>
<statement>
<p>
What is the global minimum of <m>f(x)</m>?
<ol marker= "A." cols="1">
<li> <m>1</m> </li>
<li> <m>0</m> </li>
<li> There is no global minimum </li></ol></p>
</statement>
<answer>
<p>
C
</p>
</answer>
</task>
</activity>

<activity xml:id="find-all-characteristics">
<introduction>
<p>
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