DETERMINING THE SYMMETRY OF A CURVE ALGEBRAICALLY

Key Concept

The curve is symmetrical about

(i) the y-axis if its equation is unaltered when x is replaced by -x.

(ii) the x-axis if its equation is unaltered when y is replaced by -y.

(iii) the origin if it is unaltered when x is replaced by -x and y is replaced by -y simultaneously.

(iv) the line y = x if its equation is unaltered when x is replaced y and y is replaced by simultaneously.

(v) the line y = -x if its equation is unaltered when x and y are replaced by -y and -x simultaneously.

In each case, determine what type of symmetry the curve has.

Problem 1 :

y = x2

Solution :

y = x2

Replace x by -x.

y = (-x)2

y = x2 (unaltered)

Replace y by -y.

-y = x2 (altered)

Replace x by -x and y by -y.

-y = (-x)2

-y = x2 (altered)

Replace x by y and y by x.

x = y2

y2 = x (altered)

Replace x by -y and y by -x.

-x = (-y)2

-x = y2

y2 = -x (altered)

Since the given equation is unaltered, when x is replaced by -x, the curve is symmetrical about the y-axis.

Problem 2 :

y2 = 2x + 3

Solution :

y2 = 2x + 3

Replace x by -x.

y2 = 2(-x) + 3

y2 = -2x + 3 (altered)

Replace y by -y.

(-y)2 = 2x + 3

y2 = 2x + 3 (unaltered)

Replace x by -x and y by -y.

(-y)2 = 2(-x) + 3

y2 = -2x + 3 (altered)

Replace x by y and y by x.

x2 = 2y + 3 (altered)

Replace x by -y and y by -x.

(-x)2 = 2(-y) + 3

x2 = -2y + 3 (altered)

Since the given equation is unaltered, when y is replaced by -y, the curve is symmetrical about the x-axis.

Problem 3 :

y3 = 2x

Solution :

y3 = 2x

Replace x by -x.

y3 = 2(-x)

y3 = -2x (altered)

Replace y by -y.

(-y)3 = 2x

-y3 = 2x (altered)

Replace x by -x and y by -y.

(-y)3 = 2(-x)

-y3 = -2x

y3 = 2x (unaltered)

Replace x by y and y by x.

x3 = 2y (altered)

Replace x by -y and y by -x.

(-x)3 = 2(-y)

-x3 = -2y

x3 = 2y (altered)

Since the given equation is unaltered, when x is replaced by -x and y is replaced by -y, the curve is symmetrical about the origin.

Problem 4 :

x2 + y= 5

Solution :

x2 + y= 5

Replace x by -x.

(-x)2 + y= 5

x2 + y= 5 (unaltered)

Replace y by -y.

x2 + (-y)= 5

x2 + y= 5 (unaltered)

Replace x by -x and y by -y.

(-x)2 + (-y)= 5

x2 + y= 5 (unaltered)

Replace x by y and y by x.

y2 + x= 5

x2 + y= 5 (unaltered)

Replace x by -y and y by -x.

(-y)2 + (-x)= 5

y2 + x= 5

x2 + y= 5 (unaltered)

The given equation is unaltered,

when x is replaced by -x

when y is replaced by -y

when x is replaced by -x and y is replaced -y

when x is replaced by and y is replaced x

when x is replaced by -y and y is replaced -x

Therefore, the curve is symmetrical about the x-axis, the y-axis, the origin, the line y = x and the line y = -x.

Problem 5 :

y = x3 + 1

Solution :

y = x3 + 1

Replace x by -x.

y = (-x)3 + 1

y = -x3 + 1 (altered)

Replace y by -y.

-y = x3 + 1 (altered)

Replace x by -x and y by -y.

-y = (-x)3 + 1

-y = -x3 + 1 (altered)

Replace x by y and y by x.

x = y3 + 1 (altered)

y3 = x - 1 (altered)

Replace x by -y and y by -x.

-x = (-y)3 + 1

-x = -y3 + 1

y3 = x + 1 (altered)

The symmetry test shows that the curve does not possess
any of the symmetry properties.

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