Shifts

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Section 1.5 Shifts

Subsection 1.5.1 Shifts of Conics

Figure 1.5.1. Shifts of a Circle

In Section 1.3, the equations I gave for conics were all centered at the origin. By understanding shifts of loci, I can write conics centered at any point. Shifts are a process of changing the equation of a locus to move the locus around the plane. Shifts are a great examples of the connection between algebra and geometry: I make an algebraic change to the equation and give a geometric interpretation of that chage.

I’ll describe these algebraic changes and their geometric implications. Let \(a\) be a positive real number.

If I replace all instance of \(x\) by \(x-a\) in the equation of a locus, I move the locus \(a\) units in the positive \(x\) direction.
If I replace all instance of \(x\) by \(x+a\) in the equation of a locus, I move the locus \(a\) units in the negative \(x\) direction.
If I replace all instance of \(y\) by \(y-a\) in the equation of a locus, I move the locus \(a\) units in the positive \(y\) direction.
If I replace all instance of \(y\) by \(y+a\) in the equation of a locus, I move the locus \(a\) units in the negative \(y\) direction.

Notice that all these shifts are counter-intuitive. Replacing with a negative produces a positive shift, and vice-versa.

Recall the cirlce of radius one is the locus of the equation \(x^2 + y^2 = 1\text{.}\) Figure 1.5.1 shows four different shifts of the unit circle. The new centre points show the counter-intuitive effects of the change in the algebra.

The locus \((x-1)^2 + (y-2)^2 = 1\) is a circle of radius 1 centered at \((1,2)\text{.}\)
The locus \((x+3)^2 + (y-3)^2 = 1\) is a circle of radius 1 centered at \((-3,3)\text{.}\)
The locus \((x+4)^2 + (y+6)^2 = 1\) is a circle of radius 1 centered at \((-4,-6)\text{.}\)
The locus \((x-5)^2 + (y+2)^2 = 1\) is a circle of radius 1 centered at \((5,-2)\text{.}\)