SAT II Math II : Symmetry

Study concepts, example questions & explanations for SAT II Math II

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Example Questions

Example Question #7 : Coordinate Geometry

Which of the following symmetries applies to the graph of the relation

\(\displaystyle \left (x- 4 \right )^{2} + y ^{2}= 18\) ?

I) Symmetry with respect to the origin

II) Symmetry with respect to the \(\displaystyle x\)-axis

III) Symmetry with respect to the \(\displaystyle y\)-axis

Possible Answers:

III only

I only

II only

I, II, and III

None of these

Correct answer:

II only

Explanation:

The relation 

\(\displaystyle (x-h)^{2} + (y - k)^{2} = r^{2}\)

is a circle with center \(\displaystyle (h,k)\) and radius \(\displaystyle r\) .

In other words, it is a circle with center at the origin, translated right \(\displaystyle h\) units and up \(\displaystyle k\) units.

\(\displaystyle \left (x- 4 \right )^{2} + y ^{2}= 18\)

or

\(\displaystyle \left (x- 4 \right )^{2} +\left ( y - 0 \right ) ^{2}= 18\)

is a circle translated right 4 units and up zero units. The upshot is that the circle moves along the \(\displaystyle x\)-axis only, and therefore is symmetric with respect to the \(\displaystyle x\)-axis, but not the \(\displaystyle y\)-axis. Also, as a consequence, it is not symmetric with respect to the origin.

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