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 x 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 = x^{2}
Solution :
y = x^{2}
Replace x by -x.
y = (-x)^{2}
y = x^{2} (unaltered)
Replace y by -y.
-y = x^{2} (altered)
Replace x by -x and y by -y.
-y = (-x)^{2}
-y = x^{2} (altered)
Replace x by y and y by x.
x = y^{2}
y^{2} = x (altered)
Replace x by -y and y by -x.
-x = (-y)^{2}
-x = y^{2}
y^{2} = -x (altered)
Since the given equation is unaltered, when x is replaced by -x, the curve is symmetrical about the y-axis.
Problem 2 :
y^{2} = 2x + 3
Solution :
y^{2} = 2x + 3
Replace x by -x.
y^{2} = 2(-x) + 3
y^{2} = -2x + 3 (altered)
Replace y by -y.
(-y)^{2} = 2x + 3
y^{2} = 2x + 3 (unaltered)
Replace x by -x and y by -y.
(-y)^{2} = 2(-x) + 3
y^{2} = -2x + 3 (altered)
Replace x by y and y by x.
x^{2} = 2y + 3 (altered)
Replace x by -y and y by -x.
(-x)^{2} = 2(-y) + 3
x^{2} = -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 :
y^{3} = 2x
Solution :
y^{3} = 2x
Replace x by -x.
y^{3} = 2(-x)
y^{3} = -2x (altered)
Replace y by -y.
(-y)^{3} = 2x
-y^{3} = 2x (altered)
Replace x by -x and y by -y.
(-y)^{3} = 2(-x)
-y^{3} = -2x
y^{3} = 2x (unaltered)
Replace x by y and y by x.
x^{3} = 2y (altered)
Replace x by -y and y by -x.
(-x)^{3} = 2(-y)
-x^{3} = -2y
x^{3} = 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 :
x^{2} + y^{2 }= 5
Solution :
x^{2} + y^{2 }= 5
Replace x by -x.
(-x)^{2} + y^{2 }= 5
x^{2} + y^{2 }= 5 (unaltered)
Replace y by -y.
x^{2} + (-y)^{2 }= 5
x^{2} + y^{2 }= 5 (unaltered)
Replace x by -x and y by -y.
(-x)^{2} + (-y)^{2 }= 5
x^{2} + y^{2 }= 5 (unaltered)
Replace x by y and y by x.
y^{2} + x^{2 }= 5
x^{2} + y^{2 }= 5 (unaltered)
Replace x by -y and y by -x.
(-y)^{2} + (-x)^{2 }= 5
y^{2} + x^{2 }= 5
x^{2} + y^{2 }= 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^{ }= x^{3} + 1
Solution :
y^{ }= x^{3} + 1
Replace x by -x.
y^{ }= (-x)^{3} + 1
y^{ }= -x^{3} + 1 (altered)
Replace y by -y.
-y^{ }= x^{3} + 1 (altered)
Replace x by -x and y by -y.
-y^{ }= (-x)^{3} + 1
-y^{ }= -x^{3} + 1 (altered)
Replace x by y and y by x.
x^{ }= y^{3} + 1 (altered)
y^{3} = x - 1 (altered)
Replace x by -y and y by -x.
-x^{ }= (-y)^{3} + 1
-x^{ }= -y^{3} + 1
y^{3} = x + 1 (altered)
The symmetry test shows that the curve does not possess
any of the symmetry properties.
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