// Numbas version: exam_results_page_options {"name": "Graphs: Two random transformations", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["selector", "vsh", "hsh", "vsc", "hsc", "yo", "yn", "xo", "xn", "yo0", "yo1", "yo2", "yo3", "yo4", "maxx", "maxy", "recip"], "name": "Graphs: Two random transformations", "tags": ["functions", "graphing", "graphs", "JSXgraph", "Jsxgraph", "jsxgraph", "transformations"], "advice": "", "rulesets": {"std": ["all", "fractionNumbers"]}, "parts": [{"stepsPenalty": "10", "prompt": "
\n

The point $A$ was $(-2,\\var{yo0})$ but it is now $\\big($[[0]],[[1]]$\\big)$.
The point $B$ was $(-1,\\var{yo1})$ but it is now $\\big($[[2]],[[3]]$\\big)$.
The point $C$ was $(0,\\var{yo2})$ but it is now $\\big($[[4]],[[5]]$\\big)$.
The point $D$ was $(1,\\var{yo3})$ but it is now $\\big($[[6]],[[7]]$\\big)$.
The point $E$ was $(2,\\var{yo4})$ but it is now $\\big($[[8]],[[9]]$\\big)$.

Suppose $x$ is substituted into the function $y=\\simplify[fractionNumbers,all]{{vsc}f({hsc}x+{hsh})+{vsh}}$. The order of operations tells us what happens to $x$, in our case the order is, $\\var{hsc}$ multiplies it , $\\var{hsh}$ is added to it, $f$ is taken of it, $\\var{vsc}$ multiplies it , $\\var{vsh}$ is added to it. In this case there are two transformations to the original graph.

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The $\\simplify[fractionNumbers,all]{f({hsc}x)}$ part of the equation means that to get the same $y$ value as the original graph the new $x$ value will have to be $2$ times $-2$ times the negative of half of the negative half of what it was before (so that when you multiply the new $x$ value by $\\var{hsc}$ you get the old one).

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Since the $x$ value is displayed in the horizontal direction, this means we stretch or scale horizontally by a factor of $\\simplify[fractionNumbers,all]{{recip}}$, or in other words, compress horizontally by a factor of $\\simplify[fractionNumbers,all]{{hsc}}$.

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The $\\simplify[fractionNumbers,all]{f(x+{hsh})}$ means that to get the same $y$ value as the original graph the new $x$ value will have to be $\\var{abs(hsh)}$ units less greater than the original $x$ value (so that when you add $\\var{hsh}$ to subtract $\\var{abs(hsh)}$ from the new $x$ value you get the old one).

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Since the $x$ value is displayed in the horizontal direction, this means we shift horizontally $\\var{abs(hsh)}$ units to the left right.

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Notice that $x$ is modified twice before the function $f$ gets to it. If we want the same $y$ value as the original graph we need to feed $f$ with the same input. Let $x_o$ be the old $x$ value and $x_n$ be the new $x$ value. To get the same $y$ values we want $f(x_o)=\\simplify[fractionNumbers, unitFactor, basic]{f({hsc}x_n+{hsh})}$, that is, we require $x_o=\\simplify[fractionNumbers, unitFactor, basic]{{hsc}x_n+{hsh}}$. Rearranging we see $x_n=\\simplify[fractionNumbers, unitFactor, basic]{{recip}*(x_o-{hsh})}$.

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Method one (shift then scale)

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The equation $x_n=\\simplify[fractionNumbers, unitFactor, basic]{{recip}*(x_o-{hsh})}$ tells us to take the old $x$ subtract $\\var{abs(hsh)}$ from it, add $\\var{abs(hsh)}$ to it, then multiply this result by $\\simplify[fractionNumbers, unitFactor, basic]{{recip}}$.

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Method two (scale then shift)

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We could rewrite $x_n=\\simplify[fractionNumbers, unitFactor, basic]{{recip}*(x_o-{hsh})}$ as $x_n=\\simplify[fractionNumbers, unitFactor, basic, simplifyFractions]{{recip}*x_o-{hsh/hsc}}$. This equation tells us to take the old $x$ value, multiply it by $\\simplify[fractionNumbers, unitFactor, basic]{{recip}}$, then subtract $\\simplify[fractionNumbers, unitFactor, basic]{{abs(hsh/hsc)}}$ from this result. add $\\simplify[fractionNumbers, unitFactor, basic]{{abs(hsh/hsc)}}$ to this result.

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The $\\simplify[fractionNumbers,all]{{vsc}f(x)}$ part of the equation means that the $y$ value of each point on the graph will be $\\var{vsc}$ times what they were before.  half of what they were before. a negative half of what they were before. the negative of what it was before. Since the $y$ value is displayed in the vertical direction, this means we stretch or scale vertically by a factor of $\\var{vsc}$.

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The $\\simplify[fractionNumbers,all]{f(x)+{vsh}}$ part of the equation means that the $y$ value of each point on the graph will be $\\var{abs(vsh)}$ units greater less than they were before. Since the $y$ value is displayed in the vertical direction, this means we shift vertically by $\\var{vsh}$ units.

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The graph of a function $y=f(x)$ is shown below. Move the red points so the red curve represents $y=\\simplify[fractionNumbers,all]{{vsc}f({hsc}x+{hsh})+{vsh}}$.

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"}, "xo": {"definition": "list(-2..2)", "templateType": "anything", "group": "Ungrouped variables", "name": "xo", "description": "

original x values

"}, "yo": {"definition": "repeat(random(-5..5),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "yo", "description": "

the (random) original y values which relate to the x values

"}, "yn": {"definition": "map(vsc*y+vsh,y,yo)", "templateType": "anything", "group": "Ungrouped variables", "name": "yn", "description": "

new y values after the transformation

"}, "hsh": {"definition": "if(selector[1]=1,random(-3..3 except 0),0)", "templateType": "anything", "group": "Ungrouped variables", "name": "hsh", "description": "

horizontal shift

"}, "selector": {"definition": "shuffle([0,0,1,1])", "templateType": "anything", "group": "Ungrouped variables", "name": "selector", "description": "

order is ['vsh','hsh','vsc','hsc'] 1 is on 0 is off

"}, "vsc": {"definition": "if(selector[2]=1,random(-2,-1,-0.5,0.5,2),1)", "templateType": "anything", "group": "Ungrouped variables", "name": "vsc", "description": ""}, "vsh": {"definition": "if(selector[0]=1,random(-3..3#0.5 except 0),0)\n", "templateType": "anything", "group": "Ungrouped variables", "name": "vsh", "description": "

vertical shift

"}, "yo4": {"definition": "yo[4]", "templateType": "anything", "group": "Ungrouped variables", "name": "yo4", "description": ""}, "yo3": {"definition": "yo[3]", "templateType": "anything", "group": "Ungrouped variables", "name": "yo3", "description": ""}, "yo2": {"definition": "yo[2]", "templateType": "anything", "group": "Ungrouped variables", "name": "yo2", "description": ""}, "yo1": {"definition": "yo[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "yo1", "description": ""}, "yo0": {"definition": "yo[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "yo0", "description": ""}}, "metadata": {"notes": "", "description": "

Horizontal and vertical shifts and scales of a random cubic spline

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