Activity A.1.1. Translation of Coordinates.
In this activity we will look at how to translate coordinate systems, which means to use a new coordinate system with axes parallel to to the original set of axes and the same scale. In other words, the new coordinate system has a new origin but does not change the way coordinates are measured. This is useful to simplify the equation of a graph by making the graph’s center the new origin.
(a)
On the axes below, draw and label the point \(P=(4,-1)\text{.}\) With a different color, draw a new set of axes that are centered at \(P\) and label these axes \(x^*\) and \(y^*\text{.}\) The \(x^*\)- and \(y^*\)-axes should parallel to the \(x\)- and \(y\)-axes and use the same scale.
(b)
Use the plot of the \(x^*\) and \(y^*\) axes above to give the \((x^*,y^*)\)-coordinates for each of the following points:
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\(\displaystyle (x,y)=(4,-1)\)
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\(\displaystyle (x,y)=(0,0)\)
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\(\displaystyle (x,y)=(3,-2)\)
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\(\displaystyle (x,y)=(6,3)\)
Remember that the locations of these points should not change, but rather the values of the measurements used to describe the location will change.
(c)
Generalize your work for part b and write \(x^*\) and \(y^*\) in terms of \(x\text{,}\) \(y\text{,}\) and the coordinates of new center \((4,-1)\text{.}\)
\begin{align*}
x^*\amp = \fillinmath{XXXXXXXXX} \amp y^*\amp=\fillinmath{XXXXXXXXX}
\end{align*}
(d)
Using the same coordinate systems as in part a, give the \((x,y)\)-coordinates of the following points:
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\(\displaystyle (x^*,y^*)=(0,0)\)
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\(\displaystyle (x^*,y^*)=(4,-1)\)
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\(\displaystyle (x^*,y^*)=(-5,7)\)
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\(\displaystyle (x^*,y^*)=(6,3)\)
(e)
Generalize your work for part d and write \(x\) and \(y\) in terms of \(x\text{,}\) \(y\text{,}\) and the coordinates of new center \((4,-1)\text{.}\)
\begin{align*}
x\amp = \fillinmath{XXXXXXXXX} \amp y\amp=\fillinmath{XXXXXXXXX}
\end{align*}
