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<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Hello,</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">In high school math the student is asked to graph an equation.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">In one lesson of math the student is then asked to update the equation so that the visual graph is shifted horizontally to the right by "h" and vertically up by "v".</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">As a blind high school student I found this math lesson quite challenging when I was first introduced to the subject.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">I feel this was in part because I had such limited access to a variety of tactile graphs including braille symbols showing all the ways things might work out.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">I believe that the equations listed below and their tactile representations will well prepare a blind individual when studying horizontal and vertical shifts of an equation.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">I point out that knowing about horizontal and vertical shifts of an equation remains an important concept for professionals working in the STEM fields.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">You can read the text that follows as a blind person and be well-versed in horizontal and vertical shifts of an equation.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">I believe it is quite helpful in locking down the concept to review the accompanying tactile graphics.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">It may also help to use sonification to review plots of the comma delimited CSV files.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">If I had the opportunity to review all these tactile graphics at the time I was introduced to the lesson I wonder if I would have recalled which way the shift goes for years to come without
having to generate a dataset to verify my understanding.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">You can explore the attached PNG files by placing them in a folder and using JAWS with Picture Smart.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">You can, of course, ignore all the attached files if you like.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;"> </p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">For picture Smart the JAWS keystroke is insert-space followed by p followed by f.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">You give this keystroke sequence when the PNG file is in focus in a particular folder.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">You can close the Picture Smart results window by pressing escape when finished.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">I noticed I get better results from Picture Smart and Claude</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">or chatGPT by issuing the prompt</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">"You are a high school math teacher. Please describe the graph.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Another prompt i use is</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Describe the dotted lines.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Another prompt to try is</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Describe which direction the dotted lines are going.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Another prompt is</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Are there any curves on the graph?</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">For this last prompt the chat bots have been correct each time thus far.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The chat bots give a pretty good description for Equation 4 listed below. It seems that the chat bots may be describing a common shift of the picture provided rather than the actual graph
itself.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The chat bots will not get a hundred percent on a homework problem analyzing an uploaded graph of an equation. That is one thing for sure.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;"> </p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;"> </p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">If an equation is a linear expression in the form of</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Equation 1</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">y equals m times x + b</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">then a shift by (h, v) can be achieved by the new expression or transformation</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Equation 2</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">y minus v equals m times the quantity x minus h end quantity + b</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">or mathematically</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">y-v = m(x-h) + b</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">This can be rewritten</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Equation 3</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">y equals m times x minus m times h + v + b</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">or mathematically</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">y = m(x) - mh + v + b</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Comparing Equation 3 to Equation 1 we can see that Equation 3 has a positive v on the right whereas Equation 1 does not.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">And so Equation 3 shows a vertical shift of v compared to Equation 1.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;"> </p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">In what follows I refer to a test number in particular tests 42-47.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">These tests correspond to comma delimited CSV files and corresponding generated plots for examples showing horizontal and vertical shifts of an equation.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Suppose an equation involves x and y and you wish to generate a new equation shifted to the right by h and vertically up by v.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">In this case you can transform the equation by replacing each occurrence of y with the quantity y minus v end quantity and each occurrence of x with the quantity x minus h end quantity.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;"> </p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">In test 42 I generated the graph</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Equation 4</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">y equals absolute value of 2 times x end absolute value</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">or mathematically</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">y = abs(2x)</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">This is an even function. </p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph forms the shape of a v with the vertex at the origin.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The vertical line test states that if a graph is a function then the graph should cross a vertical line no more than 1 time.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph in Equation 4 passes the vertical line test so it is a function.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;"> </p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">In test 43 I generated the graph of the equation</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Equation 5</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">x equals absolute value of 2 times y end absolute value</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">or mathematically</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">x = abs(2y)</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph of Equation 5 is a clockwise rotation by 90 degrees of the graph of Equation 4.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph is a v with the vertex at the origin. The two lines of the v go up and to the right from the vertex and down and to the</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">right from the vertex.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph fails the vertical line test.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">For example the graph contains the points (2, -1) and (2, 1).</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">These two points contain the same x but different y values and so form a vertical line segment.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The slope of the vertical line segment is the change in y which is 2 divided by the change in x which is 0.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The slope is infinite and an infinite or negative infinite slope indicate a vertical line segment.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;"> </p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">In test 44 I generated the graph of</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Equation 6</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">x equals the absolute value of 2 times the quantity y minus 2 end quantity end absolute value</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">or mathematically</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">x = abs(2(y-2))</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph of Equation 6 forms a v with its vertex at (0, 2).</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The lines of the v go up and to the right and down and to the right.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph of Equation 6 is the same as the graph of Equation 5 but shifted up by 2.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">See that this graph has the y-value of the vertex 2 where as the graph for equation 5 has the y-value of 0 for the vertex.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;"> </p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">In test 45 I generated the graph of</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Equation 7</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">x = the absolute value of 2 times y end absolute value + 2</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">or mathematically</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">x = abs(2y)+2</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph is a horizontal v with the vertex at (2, 0).</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The lines of the v go up and to the right and down and to the right.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph of Equation 7 fails the vertical line test so Equation 7 is not a function.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph of Equation 7 is the same as the graph of Equation 5 but shifted right by 2.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">See that this graph has the x-value of the vertex 2 where as the graph for equation 5 has the x-value of 0 for the vertex.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;"> </p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">In test 46 I generated the graph of</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Equation 8</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">y equals absolute value of 2 times the quantity x - 2 end quantity end absolute value</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">or mathematically</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">y = abs(2(x-2))</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph is in the shape of a v with the vertex at (2, 0).</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The two lines of the v go up and to the left and up and to the right from the vertex.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph of Equation 8 passes the vertical line test so Equation 8 is a function.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph of Equation 8 is the same as the graph of Equation 4 but shifted right by 2.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">See that this graph has the x-value of the vertex 2 where as the graph for equation 4 has the x-value of 0 for the vertex.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;"> </p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">In test 47 I generated the graph of</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">Equation 9</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">y equals the absolute value of 2 times x end absolute value - 2</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">or mathematically</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">y = abs(2x)-2</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph is in the shape of a v with the vertex at (0, -2).</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The two lines of the v go up and to the left and up and to the right from the vertex.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph of Equation 9 passes the vertical line test so Equation 9 is a function.</p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">The graph of Equation 9 is the same as the graph of Equation 4 but shifted down by 2.</p>
<p class="elementToProof" style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">
See that this graph has the y-value of the vertex -2 where as the graph for equation 4 has the y-value of 0 for the vertex.</p>
<p class="elementToProof" style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">
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<p class="elementToProof" style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">
<span style="color: rgb(0, 0, 0);">Very Best,</span></p>
<p class="elementToProof" style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;">
<span style="color: rgb(0, 0, 0);">John</span></p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;"> </p>
<p style="margin: 0in; font-family: Aptos, sans-serif; font-size: 12pt;"> </p>
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