TEST FOR SYMMTERY WITH RESPECT TO EACH AXIS AND  TO THE ORIGIN

Test for symmetry with respect to each axis and to the origin.

Problem 1 :

y = x² - 6

Solution :

Checking if it is symmetric about y-axis :

Put x = -x

y = (-x)² - 6

y = x² - 6

Hence, it is symmetric with respect to the y- axis.

Problem 2 :

y = 9x - x²

Solution :

Checking if it is symmetric about x-axis :

Put y = -y

-y = 9x - x²

y = -9x + x²

Since the equation is changed, it is not symmetric about x-axis.

Checking if it is symmetric about y-axis :

Put x = -x

y = 9(-x) - (-x)²

y = -9x - x²

It is not symmetric to the y axis.

Checking if it is symmetric about origin :

Put x = -x and y = -y

-y = 9(-x) - (-x)²

-y = -9x - x²

y = 9x + x²

It is not symmetric to the origin. So, it is neither.

Problem 3 :

y² = x³ - 8x

Solution :

Checking if it is symmetric about y-axis :

Put x = -x

y² = (-x)³ - 8(-x)

y² = -x³ + 8x

It is not symmetric to the y axis.

Checking if it is symmetric about x-axis :

Put y = -y

y² = x³ - 8x

(-y)² = x³ - 8x

y = x³ - 8x

Hence, it is symmetric with respect to the x- axis.

Problem 4 :

y = x³ + x

Solution :

Checking if it is symmetric about y-axis :

Put x = -x

y = (-x)³ - x

y = -x³ - x

It is not symmetric to the y axis.

Checking if it is symmetric about x-axis :

Put y = -y

y = -x³ - x

It is not symmetric to the x axis.

Checking if it is symmetric about origin :

Put x = -x, y = -y

-y = (-x)³ + (-x)

-y = (-x)³ + (-x)

-y = -x³ - x

y = x³ + x

Hence, it is symmetric with respect to the origin.

Problem 5 :

xy = 4

Solution :

Checking if it is symmetric about y-axis :

Put x = -x

(-x)y = 4

-xy = 4

It is not symmetric to the y axis.

Checking if it is symmetric about x-axis :

Put y = -y

x(-y) = 4

-xy = 4

It is not symmetric to the x axis.

Checking if it is symmetric about y-axis :

Put x = -x and y = -y

(-x)(-y) = 4

xy = 4

Hence, it is symmetric with respect to the origin.

Problem 6 :

xy² = -10

Solution :

Checking if it is symmetric about y-axis :

Put x = -x

(-x)y² = -10

-xy² = -10

It is not symmetric to the y axis.

Checking if it is symmetric about x-axis :

Put y = -y

x(-y)² = -10

xy² = -10

Hence, it is symmetric with respect to the x axis.

Problem 7 :

y = 4 - √x + 3

Solution :

Checking if it is symmetric about y-axis :

Put x = -x

y = 4 - √-x + 3

It is not symmetric to the y axis.

Checking if it is symmetric about x-axis :

Put y = -y

-y = 4 - √x + 3

y = -4 + √x + 3

It is not symmetric to the x axis.

Checking if it is symmetric about origin :

Put x = -x and y = -y

-y = 4 - √-x + 3

y = -4 + √-x + 3

It is not symmetric to the origin. It is not symmetry.

Problem 8 :

xy - √4 - x² = 0

Solution :

Checking if it is symmetric about y-axis :

Put x = -x

(-x)y - √4 - (-x)² = 0

-xy - √4 - x² = 0

It is not symmetric to the y axis.

Checking if it is symmetric about x-axis :

Put y = -y

x(-y) - √4 - x² = 0

-xy - √4 - x² = 0

It is not symmetric to the x axis.

Checking if it is symmetric about origin :

Put x = -x and y = -y

xy - √4 - x² = 0

(-x)(-y) - √4 - (-x)² = 0

xy - √4 - x² = 0

Hence, it is symmetric with respect to the origin.

Problem 9 :

y = x / x² + 1

Solution :

Checking if it is symmetric about y-axis :

Put x = -x

y = (-x) / (-x)² + 1

y = -x / x² + 1

It is not symmetric to the y axis.

Checking if it is symmetric about x-axis :

Put y = -y

-y = x / x² + 1

y = -x / x² + 1

It is not symmetric to the x axis.

Checking if it is symmetric about origin :

Put x = -x and y = -y

-y = (-x) / (-x)² + 1

y = x / x² + 1

Hence, it is symmetric with respect to the origin.

Problem 10 :

y = x⁵ / 4 - x²

Solution :

Checking if it is symmetric about y-axis :

Put x = -x

y = (-x)⁵ / 4 - (-x)²

y = -x⁵ / 4 - x²

It is not symmetric to the y axis.

Checking if it is symmetric about x-axis :

Put y = -y

-y = x⁵ / 4 - x²

y = -x⁵ / 4 - x²

It is not symmetric to the x axis.

Checking if it is symmetric about origin :

Put x = -x and y = -y

-y = (-x)⁵ / 4 - (-x)²

-y = -x⁵ / 4 - x²

y = x⁵ / 4 - x²

Hence, it is symmetric with respect to the origin.

Problem 11 :

y = |x³ + x|

Solution :

Checking if it is symmetric about y-axis :

Put x = -x

y = |(-x)³ + (-x)|

y = |-x³ - x|

It is not symmetric to the y axis.

Checking if it is symmetric about x-axis :

Put y = -y

y = |x³ + x|

-y = |x³ + x|

y = -|x³ + x|

It is not symmetric to the x axis.

Checking if it is symmetric about origin :

Put x = -x and y = -y

-y = |(-x)³ + (-x)|

-y = |-x³ - x|

-y = |-(x³ + x)|

-y = |x³ + x|

y = -|x³ + x|

It is not symmetric to the origin.

Problem 12 :

|y| - x = 3

Solution :

Checking if it is symmetric about y-axis :

Put x = -x

|y| - (-x) = 3

|y| + x = 3

It is not symmetric to the y axis.

Checking if it is symmetric about x-axis :

Put y = -y

|-y| - x = 3

|y| - x = 3

Hence, it is symmetric with respect to the x axis.