DETERMINING THE SYMMETRY OF A CURVE ALGEBRAICALLY

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 + y² = 5

Solution :

x² + y² = 5

Replace x by -x.

(-x)² + y² = 5

x² + y² = 5 (unaltered)

Replace y by -y.

x² + (-y)² = 5

x² + y² = 5 (unaltered)

Replace x by -x and y by -y.

(-x)² + (-y)² = 5

x² + y² = 5 (unaltered)

Replace x by y and y by x.

y² + x² = 5

x² + y² = 5 (unaltered)

Replace x by -y and y by -x.

(-y)² + (-x)² = 5

y² + x² = 5

x² + y² = 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 y 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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